G r o u p D u a l i t y a n d it s A pplications in N u c l e a r P h y s ic s
by
Santo D’Agostino
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto
Copyright © 2005 by Santo DAgostino
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract
Group Duality and its Applications in Nuclear Physics
Santo D ’Agostino Doctor of Philosophy Graduate Department of Physics University of Toronto 2005
This thesis presents a physical perspective on group duality. We explain in detail some
of the most important dualities that are of interest in nuclear physics, the relationships
among these dualities, and how the theory of dual pairs facilitates relevant physical
calculations in the context of nuclear physics. The dualities are classified according to
whether they apply to fermion systems, boson systems, or both. The role of duality in
the determination of tensor products, branching rules, and physically acceptable product
wave functions — three fundamental operations in the physical application of symmetry
analysis — is explained.
We also explain the various conceptions of plethysm (one of which is the symmetrized
product of representations), and how plethysm is related to duality. Applications of
plethysm are discussed, prim arily in the context of nuclear physics. A new method
for the calculation of plethysms and the associated algorithm, are described. This new
method has been implemented via a MAPLE program, so that plethysms (as well as the
other standard operations involving Schur functions) can now be computed interactively
in a MAPLE environment.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication
This thesis is dedicated to my mother Lucia D ’Agostino and to the memory of my father Francesco D ’Agostino.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements
I am deeply grateful to my supervisor David Rowe for his patient guidance of my research, for his kindness and generosity, for his wise and helpful advice, for his unfailing encouragement, and for teaching me many things (not just about physics). It has been a great pleasure to know him and a privilege to work with him. Thank you to Joe Repka for serving on my thesis committee, for generously giving me many hours of his gentle instruction, and for his encouragement. I thank Michael Luke for serving on my thesis committee and for his timely advice. My sincere thanks to Juliana Carvalho for suggesting that I work with David, for her many hours of patient instruction, for providing me with computer hardware and software and computer advice, and for her encouragement. I am very grateful for having had the opportunity to collaborate w ith her in research. I would also like to thank David, Joe, and Juliana for their careful reading of my thesis and for their many corrections and helpful suggestions. I am grateful to the following people for sending me papers that I was otherwise unable to obtain: C. Carre, E. Chacon, C. Cummins, A. Frank, R. Howe, M. Moshinsky, J.-Y. Thibon, and T. Ton That. It is a pleasure to thank Chairul Bahri, Stephen Bartlett, Hubert de Guise, and Peter Turner for many congenial discussions about physics and lots of other things. I have learned a lot from all of them. Thanks to Marianne Khurana for her excellent advice. Thanks to Greg Kenning for encouraging me to start the Ph.D. programme, and to the late Pat McGhie for encouraging me to finish it. I am grateful for financial support from the following sources: David Rowe, Juliana Carvalho, the University of Toronto (University of Toronto Open Doctoral Fellowship, Edward C. Stevens awards, Van Kranendonk award, bursary), the W alter C. Sumner Foundation (Sumner fellowships), NSERC (postgraduate scholarship), and the Ontario government (OGSST Scholarship). Many thanks to my mother Lucia D’Agostino and to my parents-in-law Matteo and Concetta Cirocco for their ongoing love and support. Finally, a giant thank-you to my wife Grace and our children Jasper and Kajsa. During the completion of this thesis, Jasper and Kajsa made remarkable strides in their own development, which left me both in awe and inspired. And Grace endured a difficult situation for years while I indulged myself with this project. Grace, you have my eternal and whole-hearted gratitude for your singular patience and love.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents
1 Introduction 1
2 Schur-Weyl duality 11
2 . 1 The Schur-Weyl duality theorem and its physical interpretation ...... 11 2.2 Tensor products in unitary groups ...... 15 2.2.1 Products of wave functions describing different a ttrib u te s ...... 18 2.3 P lethysm ...... 18 2.4 Branching in unitary groups ...... 19 2.5 Tensor products and branching rules for other groups ...... 20
2.6 C haracters ...... 2 1 2.6.1 Dual characters ...... 23
3 U(m) x U(n) Duality 27 3.1 The Unitary-Unitary Duality Theorems ...... 27 3.2 Character Relations for Unitary-Unitary Duality ...... 30 3.3 Common Highest-Weight States ...... 32 3.3.1 Boson C a s e ...... 32 3.3.2 Fermion C a s e ...... 34 3.4 Classification of Shell Model States: A Physical Application of Schur-Weyl Duality and Unitary-Unitary D u a lity ...... 36 3.4.1 j j c o u p lin g ...... 38 3.4.2 L — ST c o u p lin g ...... 38
4 Plethysm 40 4.1 Symmetric functions ...... 41 4.2 The plethysm concept ...... 42 4.2.1 Algebraic properties of plethysm ...... 46 4.3 A new algorithm for p le th y s m ...... 50 4.4 Plethysm for other compact groups ...... 54 4.5 Applications of plethysm in the nuclear shell m o d e l ...... 55
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.6 Appendix: A derivation of equation (4.26) using character theory .... 59
5 USp(2D) x USp(2o;) Duality 61 5.1 Unitary-symplectic operators and fermion-pair algebras ...... 61 5.1.1 The use of quasispin and seniority in pairing problems ...... 61 5.1.2 Generalizations of quasispin and seniority to groups of higher rank 64 5.2 The USp(2D) x USp(2u;) duality theorem ...... 67 5.3 Example: USp(4) x USp(4) duality on the space of the 0(16) spin repre
sentation ...... 6 8 5.3.1 Decomposition of H according to d u a lity ...... 71 5.3.2 A simple proof of the duality theorem ...... 74 5.4 Applications of USp(2f2) x USp(2w) d u a lity ...... 77
6 Sp(n,R) x 0(A) duality 80 6.1 Symplectic and orthogonal algebras ...... 81 6.1.1 sp(n, R) a lg e b ra s ...... 83 6.1.2 o (A ) algebras ...... 84 6.1.3 Symplectic and orthogonal subalgebras of sp(nA, R ) ...... 85
6.2 The Sp(n, R) x 0(A ) duality theorem ...... 8 6 6.3 Example: Sp(l,R) x 0(3) duality on the space of the Sp(3, R) oscillator
representation ...... 8 6 6.4 Common extremal states for Sp(n, R) x 0(A ) duality ...... 90 6.5 Applications of Sp(n, R) x 0(A) d u a lity ...... 93 6.5.1 Collective and intrinsic motions in nuclear collective models . . . 94 6.5.2 Classification of states in the symplectic shell m o d e l ...... 96
7 Summary 99 7.1 A brief guide to literature on d u a lity ...... 101 7.2 Further research in group d u a lit y ...... 104
Bibliography 108
A 121 A.l Introduction ...... 121 A.2 Partitions and symmetric functions ...... 124 A. 3 Transitions Among Symmetric Functions ...... 126 A.3.1 Expansion of S-functions in terms of p-functions ...... 126 A.3.2 Expansion of S-functions in terms of m-functions ...... 127 A.3.3 Expansion of m-functions in terms of S-functions ...... 128 A.4 Products of symmetric functions ...... 129 A.4.1 Physical interpretation of S-function products ...... 129 A.4.2 Product of m-functions ...... 130
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with permission of the copyright owner. Further reproduction prohibited without permission. A.4.3 The Littlewood-Richardson coefficients ...... 132 A. 5 Plethysm of S-functions ...... 133 A.5.1 An algorithm for the plethysm of S-functions ...... 135
A . 6 Illustrative example and Concluding Rem arks ...... 136
B 145 B.l Introduction ...... 147 B.2 Transitions Among Symmetric Functions ...... 148 B.2.1 Calculation of the Kostka num bers ...... 149 B.2.2 Calculation of the inverse Kostka num bers ...... 150 B.3 The Multiplication of Symmetric Functions ...... 151 B.3.1 Multiplication and division of m -functions ...... 152 B.3.2 Outer product and division of S-functions ...... 153 B.3.3 Internal product of S-functions ...... 154 B.3.4 Outer plethysm of S-functions ...... 156 B.4 Program ...... 158
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables
4.1 Selected symmetric functions of low degree in three v a ria b le s ...... 43
5.1 Infinitesimal generators of U S p(2fl) ...... 6 6 5.2 A copy of the (1) irrep of usp(4)s found in H ...... 69 5.3 A copy of the (1) irrep of usp(4)y found in H ...... 71 5.4 The branching rule USp(4) 4- S0(3) for some low-dimensional irreps . . . 78
6.1 The 0(4) I S4 branching rule for some low-dimensional irreps ...... 97
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures
1.1 Lowest energy levels for the spherical harmonic oscillator ...... 7
5.1 Weight diagram for the (1) irrep of usp(4)s ...... 69 5.2 Weight diagram for the (1) irrep of usp(4)y ...... 70 5.3 Energy level diagram for H ...... 71 5.4 Energy level diagram for H, showing spin and isospin content ...... 74
6.1 Hilbert space that carries the oscillator representation of Sp(3, R) . . . . 8 8
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1
Introduction
This thesis presents a physical perspective on group duality . 1 We explain in detail some of the most important dualities that are of interest in nuclear physics, the relationships among these dualities, and how the theory of dual pairs facilitates relevant physical calculations in the context of nuclear physics. We also explain the various conceptions of plethysm (one of which is the symmetrized product of representations), how plethysm is related to duality, and a new method for computing plethysm. This new method has been implemented via a MAPLE program, so that plethysms (as well as the other standard operations involving Schur functions) can now be computed simply in a MAPLE environment. Before describing the phenomenon of group duality, we review some concepts of sym metry analysis that are essential to the thesis: irreducible group representation, symmetry group, dynamical group, and direct product of groups. We do this in the context of the quantal description of a single particle in a spherical harmonic oscillator potential. The Hamiltonian operator for the system is, in appropriate units,
H = —p 2 + |m w 2 r 2 (1.1) mTTv £
where r is the position and p is the corresponding momentum. The possible basis 2 wave functions for such a system have three labels ( n ,l,m ), and can be written in product form as
V w ( a o, (j)) = R„t(r)Ylm(0, 4>) (1 .2 )
1In some of the physics literature, particularly the early literature, group duality is known as group complementarity. 2That is, the wave functions ipnim(r,0,(j)) form a basis for the Hilbert space of square-integrable functions.
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Additionally, the functions R n i ( r ) form a basis for a H ilbert space (the space of radial wave functions), and so do the functions Y im { d , The collection of all the Y hn functions transform among themselves with respect to SO(3) transformations. Thus, the space spanned by the Y (rn functions is said to carry a representation 3 of SO(3). Furthermore, for each value of I, the Y lm functions with the same value of I transform among themselves with respect to SO (3) transformations. No further reduction of the space w ith respect to SO(3) is possible; i.e., the space spanned by the spherical harmonics Y lm has no proper SO(3)-invariant subspace. Therefore, we say that each subspace of the spherical harmonics spanned by the Yim for a particular value of I carries an irreducible representation (irrep ) 4 of SO (3). Since symmetry groups comprise operators that transform states to other states with the same energy, they are insufficient to describe the fu ll dynamics of a physical system. W hat is needed are operators that map states of a certain energy to states having different energy. Which operators are appropriate for the spherical harmonic oscillator? For each fixed value of /, the functions Rni(r) transform among themselves via SU(1,1) transformations. Again, no further reduction of the space with respect to SU(1,1) is possible, so each space spanned by the radial wave functions labelled by a particular value of I carries an irrep of SU(1,1). For a fixed value of I, the energy of a state depends on n, and so nontrivial SU (1,1) transformations map radial wave functions w ith a particular energy into radial wave functions with different energy. To formally define group duality, one needs the concept of the direct product of groups. The direct product of the groups SU(1, 1 ) and SO(3), symbolized SU( 1 , 1 ) x SO(3), can be thought of as the set of pairs elements of the two groups. In their action on a wave function ipnim, an element of SU( 1 , 1 ) acts on the radial wave function, and an element 3 More precisely, a representation T of a group G is a Hilbert space H together with a map of G into a group of invertible linear transformations GL(H) of H that preserves the group operations. That is, for each pair of elements gi £ G, g2 6 G, T(gig2) — T(,gi)T(.g2) and T(e) = I, where e is the identity element of G and I is the identity transformation of GL(H). If every transformation T(g) of the representation is unitary, then the representation is said to be unitary. In quantal descriptions of systems, the expectation values of physical observables, which are expressed in terms of inner products, are unchanged after symmetry transformations. Since unitary operators preserve inner products on complex Hilbert spaces, group representations that are unitary are of greatest physical interest. 4A basis for an irrep of a group, which in the case of SO (3) can be taken to be the set of Yim for a fixed value of I, is also known as a multiplet, a term that originated in atomic spectroscopy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 1. I ntroduction 3 of SO(3) acts on the spherical harmonic. The direct product group SU(1,1) x SO(3) is a subgroup of Sp(3, R); a typical element of the latter would not necessarily act on a product wave function in such a simple way. Note that the H ilbert space H spanned by the ipnim carries a representation of Sp(3, R). Note also that in their actions on H, each element of SU(1,1) commutes with each element of SO(3); briefly, one says that the groups commute. Thus, for a pair of elements, one from each group, it is irrelevant which acts on a wave function first. If the groups did not commute, one could not combine their actions in such a simple way, and one would not speak of a direct product of groups. The space El can be decomposed into subspaces El*, spanned by wavefunctions ipimn with fixed values of I, each of which carries an irrep of SU(1,1) x SO(3). Each subspace H; is invariant with respect to the Hamiltonian . 5 Because each subspace of the fu ll Hilbert space that is invariant w ith respect to SU(1,1) x SO(3) is also invariant w ith respect to the Hamiltonian, we say that SU( 1 , 1 ) x SO(3) is a dynamical group for this system. As w ith symmetry groups, it is possible for a system to have more than one dynamical group. For example, the spaces Hi can be further decomposed into subspaces Him spanned by wave functions ipimn with fixed values of both I and m. Each M.lrn is invariant with respect to the Hamiltonian and carries an irrep of SU(1,1), and so SU(1,1) is also a dynamical group. The definition of a dynamical group is consistent with the physically reasonable de sire to specify a subdynamics of a system by an irrep of an appropriate group. This allows one considerable flexibility in constructing algebraic models that describe various subdynamics of the full dynamics of a system, while still being able to make use of the methods of symmetry analysis. In particular, we shall see later in the thesis how one can use group duality as a powerful tool for understanding the dynamics of complex systems. The physical essence of group duality for the spherical harmonic oscillator is that the radial dynamics and the spherical symmetry are linked. The symmetry of the system is specified by a subspace H* of the fu ll H ilbert space El, and can be labelled by the quantum number I. The corresponding spherical harmonics are solutions to the angular Schrodinger equation for the given value of I; however, the same would be true of any system with a spherically symmetric potential. The solutions Rni of the radial Schrodinger equation, which are unique to this system, express the dynamics. The radial dynamics can also be specified by the subspaces EIj, and also labelled by the quantum number I. Thus, the symmetry and dynamics of the system are interlaced, so that a specification of one immediately constrains the other. That is, the quantum numbers specifying the orbital and radial degrees of freedom are in one-to-one correspondence. Formally, if two groups are dual to one-another relative to their actions on a common H ilbert space, then each irrep of one group is paired w ith one and only one irrep of the 5A space H is said to be invariant with respect to an operator T if for every v € H, T(v) € EL Similarly, a space is said to be invariant with respect to a set of operators if it is invariant with respect to each operator. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 1. I ntroduction 4 other group. Precisely , 6 suppose that two groups, Gi and G2, both act on a H ilbert space H, and that the actions of Gi and G2 on H commute. Then the direct product group Gi x G2 can be formed. Suppose also that H decomposes into 7 irreps of Gi x G2. The groups Gi and G2 are said to be dual on H (or equivalently, to form a dual pair on H) provided that in the decomposition of H • each irrep of G\ x G2 appears no more than once, and • each irrep of G\ is paired with a unique irrep of G2, and vice versa. The key point is that each irrep of Gi x G2 in the decomosition of HI can be identified solely by the irreps of Gi, or equivalently, solely by the irreps of G2. Another example that may help to give one a sense of the physical essence of duality involves a rotating rigid object. The principal moments of inertia of the object are purely geometrical concepts, and can be calculated geometrically. However, its principal moments of inertia determine the object’s possible dynamics when it is forced to rotate. Thus, the geometrical symmetries of the object and its dynamics are intimately related, so that specifying one constrains the other. In terms of dual pairs of groups, if a group describing a system’s geometrical symmetry is dual to a group describing its dynamical symmetry then there is a one-to-one correspondence between the geometrical symmetry and the dynamics of the system. If the object is a rigid molecule, then the symmetry group is known as a point group, which is a subgroup of SO (3). Although the example of a single particle in a spherical harmonic oscillator is rather special, as the Schrodinger equation is exactly solvable analytically, it illustrates one of the main physical applications of group duality that is relevant even in cases where the Schrodinger equation is not exactly solvable. The presence of SO(3) as a symmetry group allows one to block-diagonalize the Hamiltonian, and means that the SO(3) labels L and M are good quantum numbers that can be used to partially label states. However, it is only the presence of the dual dynamical group SU( 1 , 1 ) that allows one to determine matrix elements of relevant operators and diagonalize the Hamiltonian. Furthermore, subgroup chains involving the dynamical group provide additional labels that assist in labelling states. For example, in the case of a single particle in a spherical harmonic oscillator potential, the chain SU(1,1) D U (l) provides a single additional label (the energy level N) that allows a complete labelling of basis states. In addition to central force problems such as a single particle in a spherical harmonic oscillator potential, there are numerous other physical systems for which group duality is operative. However, the phenomenon of group duality is widely applicable in physics for 6 Currently there is no single universally agreed-upon definition for group duality. In Chapter 7 we summarize the various alternative definitions in the literature. 7Technically the condition required here is that H decompose into a direct sum of irreps ofG 1 xG 2. This condition is satisfied for all of the situations of interest in this thesis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 1. I ntroduction 5 other reasons: Three fundamental processes in the application of group representation theory to physics make essential use of group duality. The first process is the combination of separate systems into a single system. Suppose that each of the separate systems is described by a unitary symmetry. If duality is present in this situation, then the unitary symmetry of the combined system is related to its permutation symmetry. A simple example is a system of two spin-| particles (i.e., “addition of angular momenta” ). Suppose that the H ilbert space H for each particle is spanned by the states {ip+, 4>-}, which can be thought of as spin-up and spin-down. The Hilbert space H 2 for the combination of the two particles is spanned by four products of the spin-up and spin-down states, where the numbers 1 and 2 keep track of the particles . 8 The space H 2 decomposes according to unitary symmetry into what are known as the triplet space r f 2* and the singlet space H i11!: H2 S H<2> © ( 1 .3 ) This will be familiar to some readers in terms of Young diagrams: = C D ® E| (!•■») The triplet space is spanned by the three states ^ + ( 1 ) ^ ( 2 ) V>+(l)tM2) + V’+(2)V’-(1) (L5) il>-( l)ip-{2) and the singlet space is spanned by the single state V,+ (l)V’-(2) — '0+(2)'0_(l) (1.6) That is, the states in the subspace transform among themselves w ith respect to unitary SU(2) transformations, and the single state in the subspace r f 11* transforms into multiples of itself with respect to SU(2) transformations. Now what is remarkable is that the basis states in each subspace also have definite particle-permutation symmetry. Even more remarkably, each of the states in Eh2l has the same particle-permutation symmetry (they are symmetric with respect to particle permutation) and it is different from the particle-permutation symmetry of the state in H i11! (which is antisymmetric with respect to particle permutation). This illustrates what is called Schur-Weyl duality. 8 We use the notation H2 to stand for H®H, the tensor product of two copies of the space EL Similarly, stands for El <8> El ® ® H (N factors), the N-fold tensor power of EL Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 1. I ntroduction 6 Schur-Weyl duality links unitary symmetry and particle-permutation symmetry. A general statement of Schur-Weyl duality applies to a system of N particles. The Hilbert space for an 7V-particle system breaks up into subspaces, where the states of each subspace have a particular unitary symmetry and also a particular particle-permutation symmetry. W hat makes duality so valuable is that there is a one-to-one correspondence between these two symmetries. This allows us to use the same quantum numbers to label both symmetries. Young diagrams are frequently used in this way, to label either or both symmetries, whichever is convenient. A good way to visualize Schur-Weyl duality is as follows. In the decomposition of the Y-particle H ilbert space H * (1.7) A the basis states of each subspace can be indicated schematically by a rectangular array where each line represents one basis state. The basis states in each row transform among themselves with respect to U(n) transformations, and the basis states in each column transform among themselves with respect to SN transformations. In other words, each row carries an isomorphic copy of an irreducible representation of the unitary group U(n), and each column carries an isomorphic copy of an irreducible representation of the particle-permutation group S^. In the case of the two spin-1 particles, the three basis states of the triplet space align into one row, as each state is invariant with respect to particle permutation. There are many applications of Schur-Weyl duality in symmetry analysis, particularly in many-body systems. For example, whenever Young diagrams and their manipulations are used to perform computations involving unitary groups, Schur-Weyl duality is im plicitly being used. Schur-Weyl duality and its applications to many-body systems are discussed in Chapters 2 and 4, and throughout the thesis. The second fundamental process alluded to earlier involves the formation of product wave functions for a single system, where each factor wave function describes a different attribute of the system. An example of this is a system that has space and spin degrees Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 1. I ntroduction 7 of freedom. If the groups that are relevant for each degree of freedom are dual, then a knowledge of the irrep that describes the space degree of freedom immediately indicates the corresponding irrep that describes the spin degree of freedom, and vice versa. Thus, if the total wave function describing the system is to have a certain property (say, being symmetric, or anti-symmetric), then duality constrains the irreps describing the individ ual attributes (space and spin) that can be combined to produce a tota l wave function w ith the desired property. Since space and spin symmetry are both described by unitary groups, this is an instance of what is called unitary-unitary duality, discussed in Chapter 3 and following. There are two varieties of unitary-unitary duality, one that applies to bosons and the other to fermions. The third fundamental process alluded to earlier is the determination of branching rules; that is, the decomposition of a representation of a group into representations of a subgroup. Such branching rules are indispensable in the labelling of basis states when carrying out concrete calculations, and where there is a reduction of symmetry, as when perturbations are applied to a system. The relevance of Schur-Weyl duality for branching rules is discussed in Chapters 2 and 4. In using group-subgroup chains (such as for state-labelling), if each member of the chain has a dual partner in another chain, then one speaks of dual chains of groups . 9 The presence of such dual chains can be particularly useful in facilitating practical cal culations, and several such chains are discussed later in the thesis. For the spherical harmonic oscillator potential, the usual energy-level diagram is shown in Figure 1.1. Figure 1.1: Lowest energy levels for the spherical harmonic oscillator N = 5 ______ N = 4 ______ N = 3 ______ N = 2 ______ N = 1 ______ N = 0 ______ 1 = 0 1 = 1 1 = 2 1 = 3 1 = 4 1 = 5 Only the first few energy levels of the diagram are shown; the energy levels extend upward indefinitely. Each line in the diagram corresponds to a multiplet of states; the 9Dual chains have been called see-saw dual pairs of groups by Kudla [86] and others. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 1. I ntroduction 8 collection of all the states spans the Hilbert space for the system, which carries a repre sentation of the group Sp(3, R ) . 10 The first column of states in the diagram spans the I = 0 irrep of SU (1,1). Each individual state also spans the I = 0 irrep of SO(3). The second column of states spans what we can call the I = 1 irrep of SU(1,1) x SO(3). Each line in the second column represents three states, labelled by m — 1 , 0 ,—1 ; each such set of three states spans a copy of the I = 1 irrep of SO (3). Taking one state from each energy level, all w ith a common value of m, produces a set of states that spans the I = 1 irrep of SU( 1 , 1 ); there are three copies of this irrep in the second column of states. A similar analysis holds for each column in the diagram, and so the entire diagram displays the duality of SU(1, 1 ) x SO(3). There is another duality at play here, which we call bosonic unitary-unitary duality in Chapter 3; the entire Hilbert space can be decomposed into energy levels, where the energy of each level (i.e., the number of energy quanta, represented by N in the diagram) is a U(l) quantum number. The collection of states in each energy level spans a U(3) irrep w ith the same label. Thus, U (l) and U(3) are also dual on this H ilbert space. Once again, we have a dual pair in which one of the groups, U(3), can be regarded as a symmetry group, and the other, U (l), plays the role of a dynamical group. Finally, the 0(1) subgroup of U (l) specifies the parity 7 r of each state. The set of all positive parity states spans an irrep of Sp(3, R), and the set of all negative parity states spans an inequivalent irrep of Sp(3, R). Thus, 0(1) and Sp(3,R) are also dual on the spherical harmonic oscillator Hilbert space. These dualities are illustrated by the following chains: SU(1,1) X S0(3) u n U (l) x U(3) u n 0 ( 1 ) x Sp(3, R) This example of a single particle in a spherical harmonic oscillator potential will be extended to multi-particle situations in Chapter 6 . In descriptions of collective motion involving the symplectic shell model, the relevant dual groups are the real orthogonal groups, which are compact, and the non-compact symplectic groups. D uality is particularly useful in this situation, since it enables one to obtain dynamical information about the system (from the representation theory of a symplectic group) using the representation theory of the orthogonal groups. It is often simpler to work with representations of orthogonal groups, since they are finite dimensional. However, there are situations where it is easier to work w ith representations 10Noncompact symplectic groups Sp(n, E) are defined and discussed in Chapter 6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 1. I ntroduction 9 of symplectic groups, even though they are infinite-dimensional. D uality allows one to work with the easier group, whichever that is in a given circumstance, and to infer results about the other group. This is a major type of application of duality. Synopsis of the thesis Chapter 2 discusses Schur-Weyl duality. The first example of duality discovered, Schur-Weyl duality is a prototype for the other dualities, and can be applied to bosons or fermions. We carefully detail the group-subgroup structure for tensor products and branching rules, so that the usual Young diagram techniques for irreps of unitary groups are clearly shown to have roots in combinatorial operations within symmetric groups. The main physical applications treated in this chapter involve the group theory underlying the construction of multi-particle wave functions having particular particle-permutation symmetries from single-particle ones. There are two varieties of unitary-unitary (U-U) duality, one for bosons and one for fermions. In Chapter 3 we show how Schur-Weyl duality implies each U-U duality, and we derive each U-U duality both by using character theory and by explicitly constructing common highest-weight states in appropriate spaces. U-U duality is used extensively in discussions involving both USp-USp duality and Sp-0 duality. The main application of U-U duality discussed in Chapter 3 is the classification of many-particle states in the nuclear shell model. The following schematic diagram indicates how the dualities discussed in this thesis are related: Fermions: SW =4> U-U — * USp-USp Bosons: SW = > U-U — >■ Sp-0 The first arrows in each row of the diagram indicate that one can derive each unitary- unitary duality from Schur-Weyl duality, as is shown in Chapter 3. The second arrow in the first row of the diagram indicates that a Hilbert space on which USp-USp duality is operative decomposes into subspaces on which fermionic U-U duality is operative. The second arrow in the second row of the diagram is interpreted similarly. Among nuclear models that are submodels of the shell model, there is a class of models known as independent particle models, and since nucleons are fermions these models fit naturally into the fermion stream of dualities (the first row in the diagram above). On the other hand, collective nuclear models fit into the boson stream of dualities (the second row in the diagram above). The fermion dualities are discussed in detail in Chapter 5, and the boson dualities are discussed in Chapter 6 . Chapter 4 deals with the concept of plethysm, which is of fundamental importance for applications in many-body physics. Suppose that one has a set of single-particle wave Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 1. I ntroduction 10 functions, each describing a particle in one of its lowest non-zero energy states. The wave functions can be combined to form product wave functions that describe a multi-particle system. To determine the particle-permutation symmetry associated with each such product wave function, it suffices to use Schur-Weyl duality. However, if more complex systems are being combined, plethysm is needed to determine the particle-permutation symmetries associated with the multi-particle wave functions. Another way to say this is that Schur-Weyl duality tells us how to decompose the tensor power of a fundamental representation, but plethysm is needed to decompose the tensor power of an arbitrary representation. For this reason, plethysm is also known as symmmetrized tensor power in some of the physics literature. In Chapter 4 we carefully explain the application of Schur-Weyl duality to plethysm calculations. The practical calculation of plethysm relies on its interpretation in terms of certain branching rules. Conversely, once an effective algorithm has been obtained, the interpretation of plethysm in terms of branching rules allows one to use plethysm algorithms to determine a large number of physically relevant branching rules. Using character theory and properties of various symmetric functions, we have obtained a new method for calculating plethysms. This method is amenable to algebraic computer calculation, and has been implemented in a M APLE program. For many of the systems of interest in many-body nuclear physics, several dualities are simultaneously relevant. Schur-Weyl duality is always of relevance when there is more than one particle. One of the major goals of this thesis is to clarify how the various dualities present in each situation are related. Finally, Chapter 7 provides a summary of the thesis. We briefly review alternative definitions of duality in the literature and provide some historical details. We also briefly review the wider scope of current research on duality, and suggest some ideas for further research. A review of group duality and its physical applications is being prepared for publica tion [30]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Schur-Weyl duality 2.1 The Schur-Weyl duality theorem and its physical interpretation Two of the most important classes of groups arising in quantal descriptions of many- body systems are unitary groups and symmetric groups. Unitary transformations are the most general that preserve the structure of the H ilbert spaces used in quantum mechanics. Furthermore, the invariance of probability that follows from unitarity in the time evolution of quantum systems is a fundamental conservation law. Symmetric groups are im portant because systems of bosons (respectively, fermions) have wave functions that are symmetric (respectively, antisymmetric) with respect to particle permutations. Schur- Weyl duality, which relates the representations of unitary and symmetric groups, provides an invaluable tool for determining which unitary symmetries may describe individual particles so that many-body wave functions have the desired permutation symmetry. This task has additional complications when particles have multiple attributes such as spatial angular momentum, spin, isospin, flavour, and colour. In this chapter we briefly describe Schur-Weyl duality, emphasizing its physical meaning. A simple example of Schur-Weyl duality was given in Chapter 1 : a system of two spin- | particles. The H ilbert space for the two-particle system decomposes into subspaces on which the quantum numbers describing unitary and permutation symmetry are in one-to- one correspondence. This unique correspondence of quantum numbers for two different groups is a hallm ark of duality. To make a general statement of Schur-Weyl duality we must first review some facts 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 12 about groups of permutations . 1 The group of all permutations of a system of N identical particles is known as the symmetric group S#. The space spanned by all possible TV-fold products of single-particle wave functions carries a representation of Sat. Irreps of S;v are carried by subspaces spanned by such wave functions that all have the same symmetry type with respect to particle permutations. In the case of two particles, as in the example described previously, there are only two symmetry types: The wave functions can be either symmetric or antisymmetric with respect to the interchange of the two particle labels. This corresponds to the fact that there are two irreps of S 2 - Irreps of S;v are identified by what are known as regular partitions of N. A regular partition of AT is a list of positive integers A = (A 1 A2 ... A*) such that A 1 + A 2 H 1-A* = N and Xi > A2 > • • • > A* > 0. Each A, is said to be a part of the partition A, and so there are k parts altogether. The two irreps of S 2 are labelled by the partitions (2) and (11) respectively. As another example, the five irreps of S 4 are labelled by the partitions (4), (31), (22), (211), and (1111). Equivalently, one can use a Young diagram to represent a partition. For example, the partition (421) is represented by the Young diagram m (2 -i) □ Now we are able to make a general statement of Schur-Weyl duality. Consider a set of n single-particle wave functions that spans a H ilbert space V, which carries the defining irrep of U(n). The Hilbert space of AAparticle wave functions, VN, spanned by products of N factors of the single-particle wave functions, can be decomposed into a direct sum of subspaces W*, where A is a regular partition of AT, which have the following properties. Each W\ has a basis of states that can be placed in a rectangular q\ x r \ array, such that each of the q\ rows carries a copy of an rvdimensional U(n) irrep, and each of the r\ columns carries a copy of a ^-dimensional S n irrep. Thus, each subspace W\ has the form 1Brief introductions to symmetric groups and their representation theory can be found in [131, §5], [28], [41, §17], and [142]. A more comprehensive source is [60, §7]. 2It can be shown (See [60, §10] or [41, §18]) that irreps of U(n) are described by partitions having n parts or fewer. If IV < n, only those irreps of U(n) having N parts or fewer appear in the decomposition of the 77-particle Hilbert space. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2 . Sc h u r -W e y l d u a l i t y 13 to enclose the partitions that refer to unitary group irreps with brace brackets. Thus, the unitary irreps for the system of two spin-| particles discussed in Chapter 1 are labelled { 2 } for the triplet and { 1 1 } for the singlet. Recall the discussion of two spin-| particles in Chapter 1. The combination of three spin-1 particles into one system leads to an 8 -dimensional H ilbert space, spanned by the states ip±( 1)'0±(2)'0±(3). This space decomposes into a 4-dimensional subspace of symmetric wave functions, spanned by the (unnormalized) states ^+(l)V’+(2)V’+(3) V;+(1)V’+(2)V’-(3) + ^+(l)V>_(2)V’+(3) + V’-(l)V ’+(2)^+(3) (2-2) (1)^- (2)^- (3) + ip - (l)'0+(2)'0_ (3) + (1)^- (2)V»+ (3) ^_(1)^_(2)^_(3) and another subspace, which w ill be described shortly. The irrep of U(2) carried by the space spanned by the four states in the previous equation is labelled by the partition {3}, or equivalently by the Young diagram I- . Each of the four basis states in the space carries the 1-dimensional symmetric irrep of S 3 labelled by (3) or the same Young diagram w ith three boxes all in one row. The other 4-dimensional subspace of the three-particle Hilbert space is spanned by the (unnormalized) states |1) = 2'0+(l)'0+(2)'0_(3) — i/,+ (l)'0_(2)'0+(3) — '0_(l)'0+(2)'i/)+(3) |2) = 2'0_(l),0_(2)'0+(3) - V>+(l) The space spanned by the states |1) and |2) carries the 2-dimensional U(2) irrep {21}, as does the space spanned by the states |3) and|4). The space spanned by the states |1) and |3) carries the ( 2 1 ) irrep of S 3 , as does the space spanned by the states |2) and|4). This is indicated schematically in the following diagram, where states in each row span equivalent U(2) irreps and states in each column span equivalent S 3 irreps. |1> |2> (2.4) 13) |4> The decomposition of the full 3-particle Hilbert space V3 can be symbolized by V 3 = Y {3} (8 ) F(3 ) © V {21} Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 14 where carries the irrep {A} of U(2) and V(\) carries the irrep (A) of S 3 . It is common to write the previous equation as V 3 = {3 } where the irrep labels in equation (2.6) stand for the subspacesof V 3 in equation (2.5) that carry the irreps. If one wishes to emphasize the U(2) transformationproperties of the 3-particle system, one can rewrite equation (2.6) as {1}® {1}® {1} = {3}® 2{21} (2.7) or, equivalently, as □ ® □ ® □ = M M © 2 ’ (2 .8 ) The factor of 2 on the right sides of equations (2.8) and (2.7) derives from the fact that the irrep ( 2 1 ) of S3 is 2-dimensional. Note that there is another irrep of S3, the antisymmetric irrep, labelled by (111), or a Young diagram having three boxes in a single column . 3 Since it is impossible to form a to ta lly antisymmetric wave function by forming linear combinations of the basis states ipa(l)ip0(2)ip1(3) when there are only two possibilities for a, /?, and 7 , the irrep { 1 1 1 } does not appear in the decomposition of the three-particle Hilbert space in this example. A formal statement of the Schur-Weyl duality theorem is as follows. Denote by V an n-dimensional vector space over C that carries the defining representation of U (n), and denote by VN the Y-fold tensor power of V. Then VN decomposes as a direct sum of irreps of U(n) xS jv, V JV = 0 F < A>® V r(A) (2.9) AhIV where brace brackets denote irreps of U(n), parentheses denote irreps of S jv, A b N means that A is a regular partition of N (i.e., a Young diagram w ith N boxes), and the direct sum extends over all Young diagrams with N boxes. As mentioned earlier, in order to be a valid U(n) irrep label, the partitions A must also have n parts or fewer (i.e., the Young diagrams must have n rows or fewer). Provided that n > N, all irreps of Sat appear in the decomposition. In alternative notation, the Schur-Weyl duality theorem is { { 1 } ® (1)) ® ({1 } ® (1)) ® • • • © ({1 } © (1)} = ^ { A } © (A) (2.10) N factors Al-IV 3In general, the symmetric irreps of Sn are identified by Young diagrams having only one row, and the antisymmetric ones have only one column. All other types of Young diagrams represent irreps of “mixed” symmetry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 15 For a formal proof of the Schur-Weyl duality theorem one can consult, for example, Chapter V of [158], Chapter 5 of [152], or Chapter 6 of [49]. Applications of Schur-Weyl duality w ill be discussed throughout the thesis. An appli cation of Schur-Weyl duality outside of nuclear physics dates from the early days of the quark model, when arguments based on it provided support for the introduction of colour as a new quark degree of freedom. [152] For example, consider the to ta lly symmetric decu- plet representation of (flavour) SU(3), which is labelled by the Young diagram . One of the basis states for this irrep consists of three up quarks, which represents the A ++ particle. Thus, the flavour wave function must be symmetric with respect to parti cle permutation. The spin of this uuu state is 3/2, and since quarks are spin-| particles, the spin wave function for uuu is also symmetric. Thus, the flavour-spin product wave function is symmetric with respect to particle permutation. Quarks were expected to be fermions and therefore the A ++ was expected to have a total wave function that is antisymmetric with respect to particle permutation. This discrepancy encouraged the introduction of another degree of freedom for quarks, colour. From the character the ory of the symmetric group and Schur-Weyl duality it follows that the only way one can have antisymmetric product wave functions is if the colour irreps are themselves antisymmetric; in other words, the particles in the decuplet must be colour singlets. In physical applications of Schur-Weyl duality, it is also the linkage between operations on irreps of unitary and symmetric groups that is of practical value. We now describe this linkage, and later generalize it to other dual pairs of groups. Two fundamental processes are the decomposition of tensor products and branching rules. 2.2 Tensor products in unitary groups The Schur-Weyl duality theorem applies directly to situations where one forms iV-particle wave functions from Ar-fold products of single-particle wave functions. However, there are situations where one needs to combine an M -particle system w ith an N -particle system to produce an (M + iV)-particle system. The Hilbert space for the (M -F jV)-particle system is spanned by wave functions that are products of an M -particle wave function w ith an W-particle wave function. Assume that the M-particle (respectively, W-particle) wave functions transform among themselves with respect to U(n) transformations, and span a Hilbert space (respectively, V ^ ) that carries a specific irrep {ji} (respectively, {z/}) of U(n). It is of physical interest to know which irreps of U(n) are contained in the (M + Ar)-particle Hilbert space <8 > In other words, we would like to decompose the tensor product {p } Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 16 and therefore <8 > C V M V N = V^M+N\ Schur-Weyl duality applied to the space VM (respectively, V N) allows us to correlate the unitary irrep {/j,} (respectively, {z/}) with the particle permutation symmetry (n) (respectively, (u)). Thus, in the actions of the relevant symmetric groups on the spaces VM boxes . 4 Finally, one can then apply Schur-Weyl duality to the space V (-M+N') to identify the correlated irreps of U(n) that are present in the tensor product space ® V^u\ which is a subspace of V^M+NK We’ll state the Littlewood-Richardson rule [60, pp. 250ff] and at the same time illustrate it with an example. Suppose that we wish to combine the Young diagrams corresponding to the partitions (31) and (21). In the second diagram, label the boxes in row k with the number k,5 1 1 1 1 1 □ 2 Now, take the numbered boxes from the first row of the second Young diagram and attach them to the first Young diagram in all possible ways so that a standard Young diagram is obtained. W ith each of the resulting diagrams, repeat this process in turn by taking boxes from subsequent rows of the second diagram. Finally, delete the numbers in the boxes and sum the resulting diagrams. A t all stages in the process follow these rules: 1 . Do not place boxes having the same number in the same column. 2 . Count the numerical entries from right to left by rows, starting at the top row. At each position in the count, the number of digits j counted must be greater than or equal to the number of digits k counted, for all j < k. That is, there must be at least as many Is as 2s, at least as many Is as 3s, ..., at least as many 2s as 3s, ... . 4The Littlewood-Richardson rule was first stated in [99], published in 1934, but proofs did not begin to be published until the 1970s; some recent proofs can be found in [151] and references therein. 5This numbering is just an artifice to facilitate applying the rule; in the end the numbers will be discarded. Since the order of the Young diagrams being combined is immaterial, it is sensible to place the diagram with more boxes first and number the diagram that has fewer boxes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 17 So, after the first stage of attaching boxes in our example, we have 1 1 1 □ 1 (2.12) □ 1 1 s 1 After the second stage we have m m 1 1 1 © □ © 1 © 2 1 2 □ 1 1 1 (2.13) s 1 2 2 Now delete the numbers in the boxes to obtain (in partition notation) (31) ® (21) = (52) © (511) © (43) © 2(421) © (4111) © (331) © (322) © (3211) (2.14) The Littlewood-Richardson rule is part of the representation theory of symmetric groups. That is, equation (2.14) can be interpreted as the decomposition of the tensor products of the irreps (31) and (21) of the symmetric group S4. However, by Schur- Weyl duality, the same result applies in the realm of unitary groups as well. In practice, since the labels for unitary and symmetric group irreps in the process described above are identical, one simply applies the Littlewood-Richardson rule without any ado to determine the tensor product of unitary-group irreps. Thus, Schur-Weyl duality, together with the Littlewood-Richardson rule, allows one to transfer a question about the tensor product of unitary group irreps into the realm of symmetric groups, answer the question, and then transfer the answer back into the realm of unitary groups. Schur-Weyl duality thereby underlies the use of Young diagrams and their manipulations 6 in the context of unitary groups. There are many other ways in which Young diagrams can be used to obtain combinatorial information in the realm of symmetric groups, and via Schur-Weyl duality, corresponding information in the realm sSee, for example, [143], [60], or [48]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 18 of unitary groups can also be obtained. Some examples of physical significance, partic ularly in many-body physics, are the determination of branching rules, tensor products, plethysms (symmetrized tensor powers), coefficients of fractional parentage, isoscalar fac tors, and recoupling coefficients. The first three applications are discussed in detail in this thesis; for the latter three applications, one can consult [140], [23], [58], and references therein. There is a large number of papers discussing tensor products of irreps for groups of physical interest. The interested reader can consult, for example, [10], [153], [113], [81], and [82]. 2.2.1 Products of wave functions describing different attributes A common situation that arises in the description of multi-particle configurations in atomic and nuclear spectroscopy is the need to construct wave functions that are prod ucts, where each factor describes a different attribute of the system. For instance, to form total wave functions for a system, one might m ultiply wave functions describing space, spin, isospin, and so on. If the particles are fermions, as they are in many atomic and nuclear models, then the total wave function must be antisymmetric with respect to all particle permutations. If they are bosons, then the total wave function must be symmetric. However, the factor wave functions need not individually be symmetric or antisymmetric. A common question in practice is: Given a multiplet of wave functions that describes one of the attributes of the system, which wave functions describing the other attributes are possible, so that the total wave function has the desired symmetry? Once again, Schur-Weyl duality is the key tool in resolving this question. A more detailed discussion with specific examples of this is provided in the following chapter. 2.3 Plethysm Schur-Weyl duality deals directly with the decomposition of a space that carries a tensor power of the fundamental irrep of U(n); that is, a space spanned by A-particle wave functions that can be thought of as being formed from products of N single-particle wave functions. Physically, this is appropriate for a description of a many-body system of identical particles for which each particle is described by the fundamental irrep. However, there are many-body systems for which each particle is described by an irrep that is not the fundamental irrep. For example, one may have several particles in a valence shell in the nuclear shell model, and so each particle may carry several quanta of energy. The relevant unitary and symmetric groups are not dual on the resulting tensor power space, except in trivial cases, since in the decomposition there is not a one-to-one correspondence of unitary and symmetric irreps. In this more complicated situation, such an A-fold Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 19 tensor power of a U(n) irrep can be decomposed into a direct sum of sets of irreps, where each set corresponds to a single type of particle permutation symmetry, labelled by a partition of N. Each such set of unitary irreps that is indexed by symmetry type is referred to as a symmetrized power, or, equivalently, as a plethysm. The practical determination of plethysms in general is a non-trivial problem, and Schur-Weyl duality is used in an essential way, both for understanding the concept, and for computing plethysms. Plethysm is discussed more fully in Chapter 4. As we shall see, for compact groups there are means for determining all plethysms. For noncompact groups the situation is much more complicated. 2.4 Branching in unitary groups Consider again a set of states that transform as a m ultiplet w ith respect to some group G. It is frequently of interest to determine the transformation properties of the states with respect to a subgroup K of G. For example, one might be studying an irrep of a unitary group and need to know the angular momentum associated with each state; in this case, G = U(n) and K = SO(3). Or there could be a perturbation applied to a system, so that the symmetry of the system that is described by G is reduced to that described by the subgroup K. Even if there is no physical relevance for a subgroup chain, the chain may still be of value in classifying basis states in which calculations can be performed. In all of these cases, one needs to know how the space that carries an irrep of a group decomposes into a direct sum of subspaces that carry a set of irreps of a subgroup. Such a decomposition is known as a branching rule. For branchings from one unitary group to another, Schur-Weyl duality is essential in the determination of the branching rule. Consider the branching rule U(m + n)|U(m)xU(n) {1} 4 . {1 } x {0 } ® {0 } x {1 } (2.15) A relevant context here is a single particle that can be in either of two shells in the shell model. Thus, in physical terms, the fundamental {1} irrep of U(m +n) can be interpreted as describing a single particle that can be in any of m + n states. Upon restriction, the particle can either be in one of the m states that carry the fundamental irrep of U ( m) or in one of the n states that carry the fundamental irrep of U(n). This can be summarized by the branching rule given in the previous equation. For an arbitrary number of particles, the states carrying an A-particle irrep of U(m + n) are separated into two sets. One set of states carries P-particle irreps of U(m), and the other carries Q-particle irreps of U(n), and one considers all such decompositions consistent with P + Q = N. The question of which irreps of U(m) x U(n) are obtained upon reduction of an irrep {A} of U(m + n) is equivalent, by Schur-Weyl duality, to the question of which irreps (p) of Sp and (u) of S q are obtained upon reduction of the given irrep (A) of Sjv- By the character theory Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 20 of symmetric groups this is also equivalent to the question of which combinations of P- particle symmetries and Q-particle symmetries are compatible with the given iV-particle symmetry. But the answer to this question is given by the Littlewood-Richardson rule, as was seen in the discussion on tensor products of unitary irreps. Thus, the branching rule for an arbitrary irrep {A} of U(m + n), and the corresponding symmetric group branching, is U(m + n)|U(m)xU(n) {A} | E E rJ-M x M P + Q = N fi\-P,v\-Q SN lS p x S Q (A) I (2-16) /jh P, v\-Q where the coefficients are known as the Littlewood-Richardson coefficients. Inci dentally, the discussion in this paragraph shows that there is an intimate relationship between tensor products and branching rules for unitary groups, which is mediated by Schur-Weyl duality. Branchings for U(m-t-n) to U(m), which are frequently of interest, can be considered to be special cases of the situation described above, where Q = 0 and P = N. Schur- Weyl duality can also be applied to determine branchings for other situations, such as U(n) to SO(3), U(n) to O(n), and so forth. Our purpose here is simply to indicate how Schur-Weyl duality is relevant to branching problems, and to outline how it is used in such cases, so further discussion of specific branching rules relevant to this thesis will be deferred to the appropriate places in later chapters. Readers interested in further details about the vast number of physically relevant branching rules can consult, for instance, [29], [78], [97], [11], and [141]. 2.5 Tensor products and branching rules for other groups Schur-Weyl duality can be used along with representation theory to determine tensor products and branching rules for other groups. For the tensor product of two irreps of a compact Lie group G, one can proceed as follows [78]. Since every compact Lie group is a subgroup of some unitary group, one uses a branching rule to associate the irreps of G with those of a suitable unitary group. Then Schur-Weyl duality and the Littlewood- Richardson rule are used to effect the tensor product of the resulting representations of the unitary group. Finally, the branching rule is used again to convert the sum of irreps of the unitary group back to a sum of irreps of G. 7 7See Section 4.4 for an example. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 21 Much work has been done on determining tensor products and branching rules for non-compact Lie groups, which are more difficult than compact ones. A comprehensive discussion of this topic is beyond the scope of this thesis, and the interested reader can consult [141], [80], [59], [81], and [82] for details and further references. We mention this subject here to make the point that one of the pillars upon which the vast subject of tensor products and branching rules for Lie groups rests is Schur-Weyl duality. 2.6 Characters In the application of group representation theory to physical problems, there are many important situations where the full information provided by representations is not needed, and the characters of representations suffice. For instance, the decomposition of tensor products and the determination of branching rules can be effected much more easily using characters, and so all of the calculations that depend on these fundamental operations (such as, for example, the splitting of energy levels upon a reduction of symmetry via a perturbation) can also be performed effectively using characters. The character of a representation for a particular group element is the trace of a corresponding m atrix. That is, if G is a group, g e G, T is a representation of G carried by the ^-dimensional Hilbert space V, and |i), where i = 1 ,..., k, is a basis for V, then the character x °f the representation T is the following complex function on the group G: k XT(s) = £ M r(9)l<> (2.17) i= 1 Since the trace of an operator is independent of the basis w ith respect to which the m atrix of the operator is expressed, the character of a representation is likewise independent of the basis used for the carrier space. Furthermore, characters are constant on conjugacy classes of a group, and are thus known as class functions. For if two elements gi and g Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 22 the one in which each element of U(n) is represented by itself. Each irrep T of U(n) con structed in this way is said to be a polynomial irrep, since the matrix elements of each T(g), where g E U(n), are polynomials in the matrix elements of g. Since the conjugacy classes of U(n) are specified by aq, x2, ■.., xn, it follows that the characters of the poly nomial irreps of U(n) are polynomials in xi,x2, • • •, xn. Furthermore, there are unitary changes of basis by which the diagonal elements of a diagonal unitary m atrix are per muted. Since the trace is invariant w ith respect to such a transformation, a character of a polynomial representation of U(n) is a symmetric polynomial; i.e., one that remains in variant with respect to any permutation of the underlying variables oq, x2, . . . , a;n. These characters of unitary group representations are known as Schur functions, or S-functions for short. Traditionally, the notation {A }, where A is a regular partition, has been used to stand for either an irrep of U(n) or its character. Another widely-used notation for Schur functions is s\, and we use both notations. According to the Weyl character formula, a Schur function can be expressed as a ratio of two determinants:8 Xj+n—j n~3 (2.18) \X. where the rows and columns of the n x n determinants in the previous equation are indexed by i and j respectively. Because of its determinantal form, the definition for Schur functions is clearly symmetric with respect to any permutation of the Xi, but it may not be clear to the reader that the expression is indeed a polynomial.9 A useful combinatorial procedure for determining an explicit expression for a Schur function, which transparently yields a polynomial, is as follows.10 To determine a formula in n variables for the Schur function s\, first draw the Young diagram associated with the partition A. Then form all possible semistandard11 Young tableaux from this Young diagram, using numbers from the set {1 ,2 ,..., n}. Associate a monomial to each tableaux as follows: if the tableaux contains the digit 1 with multiplicity mi, the digit 2 with multiplicity m2, ..., the digit n with multiplicity m„, then the associated monomial is x™'x™ 2 ■ • -x™n. Sum all of the monomials and the result is an expression for the Schur function s\ in n variables. For example, to determine an expression for S(2i) in 3 variables, begin with the 8This ratio of determinants was identified and studied as far back as 1815 by Cauchy [21] and 1841 by Jacobi [72], although not in this context. 9See Littlewood [98, p 91]. 10For a short proof of the equivalence of the determinantal and combinatorial definitions of Schur functions, see [115]. 11That is, the numbers increase going downwards in each column, and they do not decrease going to the right in each row. Semistandard Young tableaux are also known as unitary numberings of Young tableaux in some of the literature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 23 following semistandard tableaux: 1 1 1 1 1 2 1 2 1 CO 1 3 2 2 2 3 (2.19) 2 3 2 3 2 3 3 3 Now sum the associated monomials to obtain S(21) x\x2 + xfxs + xixl + xix2xs + xix 2xs + xixl + 0:2 3 :3 + = y 'x%xt + 2xix2x3 (2.20) In other words, the character of an irrep of a unitary group is the sum of the weights of the weight states. Since the characters of other groups of physical interest (such as orthogonal, unitary- symplectic, and real symplectic, although for the latter infinite series of Schur functions axe required) are expressible as linear combinations of Schur functions, and vice versa, operations on Schur functions, some of which are discussed elsewhere in this thesis, can be used in the character theory of those groups too. Thus, Schur functions are of central importance in the application of character theory to the solution of physical problems. 2.6.1 Dual characters Since the irreps of unitary and symmetric groups are linked via Schur-Weyl duality, it is natural to expect the characters of unitary and symmetric groups to have some similar linkage. One can speak of the characters of symmetric and unitary groups as also being dual. To state the duality of characters of symmetric and unitary groups, we first need to introduce another class of symmetric polynomials, known as power-sum symmetric functions, and denoted by p\, where, as usual, A is a regular partition of N. If A = (r) has just one part, then the power-sum symmetric functions in n variables are defined as pr = x[ + xr2 + • ■ • + xrn (2.21) For a general partition A = (A 1 A2 ... A&), the definition of the power-sum symmetric functions is P\=P\,P\2 --'P\h (2 -2 2 ) Using Schur-Weyl duality, one can show that the characters of unitary and symmetric groups are dual, in the following sense.12 From equation (2.9), the Schur-Weyl duality theorem is U iV = 0 y < A> 12See [9, pp 12 ff) and [127]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 24 where V = C carries the fundamental irrep of U(n), and it is understood throughout this argument that the number of parts of each partition is less than n. To determine the character relation implicit in the Schur-Weyl duality relation, we determine the trace of the action of U(n) xSN on each side of equation (2.23). It suffices to restrict attention to diagonal matrices in U(n) and permutations of particular cycle-types in Sjv, since they characterize conjugacy classes. Let the basis vectors of the tensor power space VN be represented by E(ix, i2, . . . , %n) — e t -E (iu i 2,...,iN) = diag(xx,x 2,...,xn)-(eil®ei 2 ® ---® e iN) (2.24) x The previous equation shows that each basis vector E(ix,i2, • • ■, *jv) of VN is a weight vector of U(n) (that is, an eigenvector of the action of the diagonal subgroup D of U(n)). The action of an S jv permutation P on basis vectors of VN is P -E {iu i 2,...,iN)=E (P 1 (zi), P 1 (i2),... ,P 1{ i n ) ) (2.26) Thus, the action of D x on basis vectors of VN is (t x P) -E(ii, i2,. ..,iN) = xhxi2 -xiN ■ E (P l {ix ),P \ i2),...,P 1 (iN)) (2.27) The character of the D x Sjv action is then (2.28) where the sum extends over all eigenvectors of the joint action. To simplify the expression for the character, we must identify the form of the eigenvectors of the joint action. To see how to do this, suppose in itia lly that N = 3 and that P has cycle-type (21); for example, P = (12) (3). Then the character has the form (2.29) with the condition that two of the subscripts im are equal. Thus, the character is (2.30) In general, if the perm utation P has cycle-type (p) = (NVN, [N — I ]"*-1,... where ux + 2u2 + 3^3 -I 1- N vn = N, then the character has the form of equation (2.28) with the condition that the subscripts im break up into sets of Uj, where j — 1, 2,3,..., N, such that the subscripts w ith in each set are equal. Thus, the character has zqv factors of the form ^2 jN xfN, vN~i factors of the form ^ • JV_ £ , ..., p2 factors of the form Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 25 J^j2 xj2 1 and vi factors of the form Xjl . Thus, the character of the action of U(n) x Sjv obtained by determining the trace of the left side of equation (2.23), for a diagonal element diag(xi, , xn) of U(n) and an element of Sn of cycle type (u) is (2.31) To determine the trace of the right side of equation (2.23), we use the fact that each summand on the right side of equation (2.23) is a tensor product space, and hence its character is simply the product of the characters of the factor spaces. Thus, we are led to the following relation between the characters of symmetric and unitary groups, which may be taken as the character version of Schur-Weyl duality: Pp ~ / . X/j,sx (2.32) where x£ is the character of the S/v irrep labelled by A for the conjugacy class labelled by ii. A straightforward series of manipulations enables us to express Schur functions in terms of symmetric group characters and vice versa. To begin, an orthogonality relation for symmetric group characters is (2.33) where gtl is the number of elements of the conjugacy class of S^ labelled by ji. Multiplying both sides of equation (2.32) by #MxJ) and summing on /i, one obtains (2.34) (2.35) N\ su (2.36) Thus. (2.37) The previous equation expresses the characters of unitary groups (S-functions) in terms of characters of symmetric groups. A remarkable and extremely valuable feature of equa tion (2.37) is that it is valid for all U(n), and yet the value of n does not appear in the equation. It follows that the Littlewood-Richardson rule for the product of Schur functions is independent of n, and therefore the manipulations of Young diagrams for cal culations in character theory, familiar in many physical applications, are also independent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 2. Sc h u r -W e y l d u a l i t y 26 of n. Equation (2.37) can be inverted by using an inner product 13 defined on symmetric functions, w ith respect to which Schur functions of degree N form an orthonormal basis for the space of symmetric polynomials of the same degree: (s/x, sv) = 6 ^ (2.38) From equation (2.32), it follows that xZ = (p»,Su) (2.39) Equation (2.39) states that by expressing power symmetric functions in terms of linear combinations of Schur functions, one can read the characters of the symmetric groups as the coefficients. The duality of unitary and symmetric group characters allows one to define a map, known as the characteristic map of Frobenius, that associates each character of a sym metric group with the corresponding character (i.e. S-function) of a unitary group. The existence of this map is another expression of Schur-Weyl duality, and we tacitly make use of this map when applying the character theory of symmetric groups to unitary groups (for example, in using the Littlewood-Richardson rule to determine tensor products of unitary irreps). * * * In this chapter, we have outlined how tensor products and branching rules in unitary groups are related by Schur-Weyl duality to operations in the realm of symmetric groups. This connection allows one to use the considerable machinery of Young diagrams, which relate directly to symmetric groups, for efficient calculations in the realm of unitary groups. Far from being an esoteric result, Schur-Weyl duality is seen to be fundamental for the application of representation theory to physical systems. In particular, Schur-Weyl duality is of great value in the classification of states with many degrees of freedom, as occurs frequently in nuclear, atomic, and molecular physics. Further concrete examples of the application of Schur-Weyl duality w ill be discussed in the following chapters. 13This inner product is distinct from the inner product of functions on the group manifold of a unitary group, defined in terms of integration with respect to an invariant measure on the group manifold. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 U(ra) x U(n) D uality In this chapter we discuss the duality of the groups U(ra) and U(n) in their actions on a Hilbert space that carries an irrep of U(m n). There are actually two such dualities, one that is relevant for bosons and the other for fermions, so we refer to these as bosonic unitary-unitary duality and fermionic unitary-unitary duality, respectively. The main physical application for unitary-unitary duality discussed in this chapter is the construction of product wave functions where each factor describes a different attribute of a system. For m ulti-particle systems, one must also consider how the total wave function for the entire system is to be constructed from the individual wave functions for each particle. This is the subject of coupling schemes, and is discussed further in the last section of this chapter. 3.1 The Unitary-Unitary Duality Theorems Consider the construction of the wave function of a single particle that has two attributes, which could be spin and isospin, or any other two relevant attributes. One attribute is described by irreps of a unitary group U(m) and the other by irreps of U(n). The product wave functions span irreps of the group U(mn). If the particle is a boson, then a product wave function that describes it is symmetric w ith respect to particle-permutation symmetry. Thus, by Schur-Weyl duality, all possible such product wave functions span a Hilbert space that carries irreps of U (ran) whose Young diagrams have a single row; that is, the irreps {iV }, for some value of N. I f the particle is a fermion, then a product wave function that describes it must be antisymmetric, by Schur-Weyl duality. All possible such product wave functions span a Hilbert space that carries irreps of U(nra) whose Young diagrams have a single column; that is, the irreps {1^} = {111... 1}, for some 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 28 value of TV. Physically, one can interpret the fundamental relationship K iy 4 . v i ‘ > X V„(‘ > (3.1) that defines the branching U(ran) I U (m) x U(n) (3.2) as follows. The fundamental {1} irrep of U(mn) can be interpreted as describing a single particle that can be in any of the mn states that carry the fundamental irrep. Each of the groups U(m) and U(n) serve to describe different attributes of the same particle; for example, one may describe spatial degrees of freedom, and the other may describe spin degrees of freedom. Thus, the states that carry the {1} irrep of U(mn) are formed from all possible combinations of tensor products of states that carry the fundamental irreps of U(m) and U(n), so that all possible combinations of attribute values of the single particle can be accounted for. In the boson case, the groups U(m) and U(n) are dual in theiractions ona Hilbert space Vmn^ that carries the irrep {T V } of U(mn), for each value of TV.The Hilbert space decomposes according to the following relation, which we call the bosonic unitary-unitary duality theorem: U (m n )4 .U (m )x U (n ) y W | x (3.3) a where the sum extends over all partitions a of TV that have a number of parts fewer than both m and n. Clearly the correlated irreps of U(m) and U(n) are in one-to-one correspondence in the decomposition. The bosonic unitary-unitary duality theorem is derived in Section 3.2. In the fermion case, the groups U(m) and U(n) are dual in their actions on a Hilbert space Vmh * that carries the irrep { l w} of U(mn), for each value of TV. The Hilbert space decomposes according to the following relation, which we call the fermionic unitary- unitary duality theorem: U(m n) I U(m) x U(n) I £ V™ x V™ (3.4) where the sum extends over all partitions a of TV that have a number of parts fewer than both m and n, and a is the partition conjugate to a. The Young diagrams for partitions that are conjugate to one another are reflections of each other in their main diagonals, which extend downwards and to the right beginning at the left-most box in the first row. For example, (2) = (11), (21) = (21), and (531) = (32211). By construction, the conjugate of a partition is unique, and so once again the correlated irreps of U(m) and U(n) are in one-to-one correspondence in the decomposition. The fermionic unitary- unitary duality theorem is derived in Section 3.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(ra) x U(n) D u a l i t y 29 Besides the spaces discussed in the previous paragraphs, there are no other spaces of irreps of U(mn) for which U(m) and U(n) are dual. This can be shown using Schur-Weyl duality via the following argument; the argument does more, actually proving the duality theorems.1 Suppose on the contrary that U(m) and U(n) were dual on a H ilbert space that carries an irrep {A} of U(mn), where A b N. Then by Schur-Weyl duality, there would have to be a series of uniquely paired irreps (p;) and (v) of S n such that (A) is contained in the decomposition of each tensor product (p) U(mn) 4 . U(m) x U(n) (3.5) the branching has the form {A } 4 x m <3-6) where the sum extends over all partitions of N. It follows from Schur-Weyl duality that the corresponding irreps of Sat that specify the particle-permutation symmetries satisfy the restriction (At) 1It can be shown that Schur-Weyl duality and unitary-unitary duality each imply the other. For proofs see [66] or [52]. 2This follows from the total symmetry of the coefficientsk*v in the product of symmetric group characters xpXv = 2 a k»vXX- The symmetry of the coefficients can be seen by multiplying each side of the previous equation by xp and applying the orthogonality relation (2.33). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 30 3.2 Character Relations for Unitary-Unitary Duality The unitary-unitary duality theorems can be verified by a direct calculation of the cor responding branching rules using characters, as we show in this section. Before looking at the general case, consider the example of U(4) 4- U(2) x U(2). Suppose that V carries the fundamental {1} representation of U(4), V\ carries the fun damental {1} representation of one of the U(2) groups, and V2 carries the fundamental {1} representation of the other U(2) group. Then V = V\ ® V2 , and this can be written in terms of the branching rule for the representations as { 1 } 4 - { 1 } x { 1 }, or in terms of characters as S(i)(z) = S(X)[x)s(X)(y). Let diag(zx, z2, z3, Z 4) specify a conjugacy class in U(4), and let diag(xi, x2) and diag(yi, y2) specify conjugacy classes in each of the U(2) groups. Making use of the simple embedding zx xxyx, Z2 xx2/2 , 2 3 <-> x2yx, Z4 S(2 )(z) = zf + Z1Z2 + ZXZ3 + Z1Z4 + z\ + Z2Z3 + Z2Z4 + z\ + z3z4 + z\ (3.8) in terms of Schur functions of the a;-variables and y-variables. This is straightforward in this example, with the result s(2 ) (z) = xiyixiy-i + xxyixiy2 + Xiyix2yx + xxy\x2y2 + xxy2xxy2 + xxy2X2yx + xxy2x2y2 + x2yxx2yx + x2yxx2y2 + x2y2x2y2 = (xl + xxx2 + xl) (yf + 2/12/2 + 2/1) + {xxx2) (2/12/2 ) (3.9) and thus, { 2 } 4- { 2 } x { 2 } © { l 2} x { l 2} (3.10) which proves the duality theorem for this case. The same kind of argument shows that { I 2} I {2 } x { l 2} © { l 2} x {2 } (3.11) Combining the results of the previous two equations, one sees how the full tensor product space V ® V decomposes into subspaces that carry various irreps of U(2) x U(2). In terms of characters, this decomposition is [s(i) ( * ) ] 2 = tabOL) ] 2 [s(i)(2/ ) ] 2 (3.1.2) To derive the unitary-unitary dualities in general, we need to consider the subspaces y W and y*1"} of the iV-fold tensor product space V N, and determine how they de compose into spaces that carry irreps of U(m) and U(n). Since V = Vx ® y2, it follows that VN “ VXN ® V? (3.13) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 31 Now, considering the action of a diagonal matrix of U(ran) (respectively, U(m) x U(n)) on the left side (respectively, right side) of the previous relationship, and using the fact that the trace of a tensor product of matrices is the product of the traces of the matrices, it follows that h i ) (*)]N = [S(1 ) (s)]N [s(i) (y)]N (3.14) Applying Schur-Weyl duality (see equation (2.9)) to the space V N, we get [»(!)(*)]* = SXeSAW (3.15) AHA where e stands for the identity class of S#. Similar relations hold for and V f , and so it follows from equation (3.14) that Y ^ sAy) AHA Lix\-N jj\-N = Y XeXe^{x)Su(y) (3.16) (i.i'HN Applying the rule for the tensor product of symmetric group characters ApApyV yu = / ^ xrA fji/Ap y A (3.17) Ah AT to equation (3.16), we get 5 ^ X e « A h )= Y VlvXXeS„{x)sv{y) (3.18) AHA The linear independence of Sn characters implies that the previous equation reduces to sa(a) = Y Vlv^{x)su{y) (3.19) fiyl/h-N which is the branching rule for U(mn) 4- U(m) x U(n). From the character theory of symmetric groups, we have = ^ and r^) = ^ (3.20) and inserting these relations into equation (3.19) results in sw h ) = Y s»(x)s»(y) and s(iw)(^) = Y s»(x)sv(y) (3.21) lih N fil- A which proves the unitary-unitary dualities. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3. U (m) x U(n) D u a l i t y 32 3.3 Common Highest-Weight States To carry out concrete many-body calculations, one frequently needs to construct specific states in a many-body H ilbert space. Once highest-weight states are identified, all other states in the space can be constructed by applying lowering operators of the appropriate algebras. In this section we indicate how such highest-weight states are constructed. [105, 102] 3.3.1 Boson Case In the decomposition of the Hilbert space y W that carries the irrep {N} of U(m n) into subspaces y lAlxlAl that carry the irreps {A} x {A} of U(m) xU(n), each V W dM contains a single state that has highest weight with respect to both U(m) and U(n). In this section we display such common highest-weight states for the bosonic unitary-unitary duality. The fundamental irrep {1} of u(mn) is spanned by the mn states of the form 6^|0). / \ N The space y W that carries the {iV} irrep of u (mn) has highest-weight state [bhj |0). The other basis states in y W can be taken to be all the other monomials of degree N in boson creation operators acting on the vacuum. We assume that m < n. Young diagrams can be used to guide the construction of common highest-weight states. Each state in the Hilbert space corresponds to a semi-standard Young tableau, which is a numbering of the appropriate Young frame in such a way that the numbers increase down columns and do not decrease along rows. For example, to construct a state of the space y I A>x IA> that has highest weight with respect to U(m), we place as many Is as possible into the Young diagram, then place as many 2s as possible in the remaining boxes, etc. To satisfy the semi-standard constraint, the only way to accomplish this is to fill the first row of the Young diagram with Is, fill the second row with 2s, and so on. The same argument applies to U(n), and so the common highest-weight state is labelled by identical Young tableaux for both U(m) and U(n). Since the tableaux have the same shape, it is convenient to use only one diagram, and place two numbers in each box, with the understanding that the first number applies to U(m) and the second to U(n). For example, the common-highest weight state for A = (5,3, 2 ) has Young tableau 1 1 1 1 1 1 1 1 1 1 to 2 2 to to to (3.22) CO CO 33 The character theory of symmetric groups suggests that one anti-symmetrize within columns, and so using Schur-Weyl duality, one is led to propose the following state as a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 33 potential common highest-weight state corresponding to the Young tableau in (3.22): ( d i ) 2 io> (3.23) The u(m) operators act on the first numbers in the tableau’s boxes, and so to construct the Young tableaux corresponding to the remaining states of the U (m) irrep labelled by A, one keeps the second numbers in each box of the tableau constant and increases the first numbers in the boxes so that the first numbers form valid semi-standard tableau. (Of course, there are m ultiple copies of this irrep, and one obtains the m ultiple copies by altering the second numbers in the tableaux in the same valid way for each state in the irrep.) In general, the following state in y W is a highest-weight state with respect to both U(m) and U(n), with weight (Ai, A2, ..., Ar): A?'—1 —A t < CO 1 At & • °l(r—1) 4 . ■ ■ Ai — A2 b\ 1 W = ( * !,) |0> b\i i>22 At V i ) i °(r-l)(r—1 ) t>u ■ • Hr (3 where r is the number of parts of A. The Young tableau for the highest-weight state has Ar columns with r boxes each and A* — Xk+i columns with k boxes each, where k = 1,2 , . . . , r — 1. The state |A) in equation (3.24) is meaningful only if r is less than both m and n, which explains why this constraint is necessary. A straightforward calculation shows that the state |A) in the previous equation is indeed a common highest-weight state. That it is a weight state for both U(m) and U(n), with the correct weight, can be seen as follows. The U(rrc) weight operators essentially count the number of factors of b\s, as do the U(n) weight operators. Thus, one obtains the first component of the weight by summing all of the exponents in equation (3.24), the second component by summing all of the exponents but the first, the third component by summing all of the components but the first two, and so on. That it is a highest-weight state follows from the determinantal form of the state. One can try to raise the weight by decreasing certain subscripts of the creation operators appearing in the state, but that will inevitably lead to two rows or two columns of a determinant being identical, and so the action of all of the raising operators of both U(m) and U(n) on the state (3.24) annihilate it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 34 3.3.2 Fermion Case In this section we explicitly construct common highest-weight states for fermionic unitary- unitary duality. The fundamental irrep {1} of u(ran) is spanned by the mn states of the form aj.,|0). The space V ^ ^ that carries the { 1 ^ } irrep of u(m n) has a basis consisting of all states of the form c 4 p iGi 2 i2 ''' where no two factors of a) have the same subscripts. Such states are non-zero provided that N < m. (By assumption, m < n, so it follows that also N < n.) Each state • • • o^A,;!VjO) is a weight state for u(m) (respectively, u(n)) with weight (Ai, A2, ..., Ajy) (respectively, (/zi, /i2, • ■ •, Hn )), where Xs (respectively, Hi) is the number of occurrences of s (respectively, i) in the first (respectively, second) subscripts of the a) factors. Note that ^2S Xs = JT Hi = N. One can show in a relatively straightforward way that decomposes into irreps of u(m) © u(n) according to the duality relation. That is, we show that a state of the form aiih ak2h ' " alNlN I®) (3.25) is a common highest-weight state for u(m) and u(n) provided that the weights determined above, A = (Ai, A2, . . . , Ajv) and h = (i^u ^ 2, ■ ■ ■, Hn), are regular partitions of N (that is, Ai > A2 > ... > Atv, and similarly for h) and also provided that A = fi. The irreps of u(m) and u(n) that appear in the decomposition of f/iliVl must be labelled by partitions of N, since the polynomials in V ^1 1 have degree N. Consider the Young diagram for one such partition A, which specifies an irrep of u(m ). The highest weight state for this irrep can be labelled in the usual way by placing only Is in the first row of the Young diagram, only 2 s in the second row, and so on. Any state of the form (3.25) for which there are A.s occurrences of s in the first subscripts is a highest weight state of the irrep A of u(m). (Thus, each irrep of u(m) that is labelled by a partition of N occurs in the decomposition of V ,[ l b) O f these states, which has highest weight with respect to u(n)? The state that has highest weight with respect to u(n) has the most occurrences of 1 in the second subscripts; if there are more than one w ith an equal number of such occurrences, the state with highest weight has the most occurrences of 2 in the second subscripts, and so on, subject to the constraint that no two factors of a 1 have the same subscripts. For example, consider A = (6,4,3,1). The state that has highest weight with respect to u(m) is 1 1 1 1 1 1 2 2 2 2 (3.26) 3 3 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 35 The numbers in the Young tableau correspond to the first subscripts in the state of the form (3.25). The second subscripts that occur in the state of the form (3.25) can also be placed in the Young tableau. Thus each box in the Young tableau will have two entries, which are identical to the first and second subscripts of the corresponding factor of a t in the state (3.25). Considering now the set of second entries in the Young tableau, the way to include as many Is as possible, and still satisfy the constraint that no two boxes have the same entries, is to place Is in the first column: 1 1 1 1 1 1 1 21 2 2 to (3.27) CO 3 3 41 Similarly, we place 2s in the second column, 3s in the third column, and so forth, in order to create a state that has highest weight with respect to u(n). Thus, the common highest-weight state in this case corresponds to the Young tableau Il|l2ll3|l4|l5|l6 21 22 23 24 (3.28) 31 32 33 41 and has the explicit form It follows that there is only one state of the form (3.25) that is a common highest-weight state associated with the partition ( 6 ,4,3,1) with respect to u(m). In general, a common highest-weight state has the form Ai A2 Ar n«ViR-lRp> (3-30) $1 = 1 i 2 ==l ?r = 1 Making the same arguments in the general case, one concludes that within the irrep { lw} of u(mn), each irrep of u(m) labelled by a regular partition of N appears; i.e., provided that the number of parts of A is fewer than m, and by assumption, also fewer than n. Furthermore, each such irrep {A} of u(m) is correlated uniquely with the irrep {A} of u(n). Finally, by the construction of the common highest-weight states, it is clear that in the decomposition each irrep of u(m) © u(n) occurs only once. This verifies the fermionic unitary-unitary duality relation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 36 3.4 Classification of Shell Model States: A Physical Application of Schur-Weyl Duality and Unitary- Unitary Duality The main physical application for unitary-unitary duality that is discussed in this chapter is the construction of product wave functions where each factor describes a different attribute of a system. This procedure is essential in many-body physics, where wave functions for many-particle systems are constructed as linear combinations of products of single-particle wave functions. For instance, one may form a product wave function having one factor that describes spatial angular momentum (space) and the other that describes spin. The spatial wave functions carry irreps of U(m ), the spin wave functions carry irreps of U(n), and the total wave functions carry irreps of U(ran), for suitable values of m and n. In practice, one is interested in which spaces carrying irreps of U(m) and U(n) can combine to produce a desired carrier space for an irrep of U(mn), which irreps of U(m) and U(n) result from the decomposition of a particular irrep of U(rrm), and so on. U nitary-unitary duality is a useful tool in answering such questions in those situations where it is present; if unitary-unitary duality is not present, one may make use of the plethysm concept (based on duality), which is discussed in detail in the following chapter. By the manner in which many-particle wave functions are constructed, Schur- Weyl duality is always operative, and is an essential tool in the analysis of many-body systems. Numerical calculations in the shell model require the specification of a finite-dimensional model subspace of the infinite-dimensional shell model space. Once one has defined a model subspace, one must specify a useful basis and a scheme for labelling basis states, both to identify which states are relevant and to distinguish relevant states from one another. In general, a prim ary way of labelling basis states involves the use of a suitable chain of subgroups. For instance, the ordinary means for labelling states that carry irreps of unitary groups in terms of weights can be thought of in these terms (w ith the appro priate chain being U(m) D U (l) x U (l) x • ■ ■ x U (l)), as can the Gel’fand-Tsetlin scheme (with the relevant chain being U(m) D U(m — 1) D U(m — 2) D • • • D U(2) D U (l)). However, a chain of subgroups that is mathematically convenient for classifying states in a model subspace is not necessarily physically relevant. Thus, one strives to use a chain of subgroups for labelling basis states that supports useful physical interpretations. (For some physically relevant subgroup chains, the number of labels that arise is insufficient to properly label the states. This difficulty goes by the name of the state-labelling problem, and requires the construction of additional independent operators for which the basis states are eigenstates, so that the resulting eigenvalues provide the necessary labels [130].) Having obtained a labelling system for single-particle states, one then faces Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 37 the same task for many-particle states. More accurately, in practice one begins with single-particle states and faces the task of choosing a physically relevant basis of many- particle states for the model subspace; i.e., a physically relevant group-subgroup chain, plus a unique labelling scheme for basis states. That each nucleon in a nucleus has several attributes — i.e., space (orbital angular momentum), spin, and isospin — adds another element to the problem, for one must mul tip ly three different wave functions to obtain a total wave function for just one nucleon. The complete problem, then, is to combine (couple) all of the factor wave functions for each particle to obtain a total multi-particle wave function. The solution is that this can be done in a number of different ways, and depending on the physical circumstances, one coupling scheme w ill be more useful than others. In this section we focus on j j coupling and L — ST coupling, detailing the group-subgroup chains that are relevant for each, and explaining how duality is a helpful tool. The final consideration in the construction of multi-particle nuclear wave functions is the Pauli principle. For models based on fermions (as opposed to models that treat cou pled pairs of nucleons as bosons), a total wave function for the entire multi-particle sys tem must be antisymmetric with respect to particle permutations, according to the Pauli principle. The Pauli principle does not place restrictions on the particle-permutation symmetries of the factor wave functions. If some factor wave functions have particular symmetries, then one needs to know the compatible symmetries for the other factors so that the total wave function has the appropriate particle-permutation symmetry. Group duality provides the means for ensuring that total wave functions have the required sym metry properties. One of the fundamental assumptions of the shell model is that nucleons can be con sidered, at least in first approximation, to be relatively independent and all move in an averaged potential field, which is created by all of the nucleons together. In subsequent approximations, one then considers interactions among the particles, and also the inter action between the spatial angular momentum and the spin angular momentum for each of the particles. If the spin-orbit interaction for individual particles is relatively large, then a better physical approximation results if one first combines the wave functions describing spin and orbital angular momenta for individual particles, then couples the resulting products together to form a total wave function for the entire multi-particle system. This method is known as the j j coupling scheme. On the other hand, if it turns out that the spin-orbit interaction for each individual particle is relatively small, then a better physical approximation results if one first com bines all of the spin wave functions for each particle together to form a total spin wave function, and similarly for the spatial angular momenta, and finally combines the total spin wave function with the total spatial wave function to form a total wave function for the entire m ulti-particle system. This method (once isospin is also included) is known as L — ST coupling. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 38 3.4.1 jj coupling A wave function describing a single nucleon has space, spin, and isospin factors. In j j coupling, wave functions describing the spatial angular momentum I, the spin s = and the isospin r — ± | are first combined to form a wave function that describes angular momentum j = / ± | , parity (—1);, and isospin r. These single-particle wave functions are then combined to form multi-particle wave functions, describing total angular momentum J, parity 7r, and isospin T. In a single j-shell, there are 2j + 1 possible values for the total angular momentum, and 2 possible values for the isospin, and so there are 2 (2 j + 1 ) single-particle basis wave functions. Thus, the single-particle basis wave functions span the irrep { 1 } of the unitary group U( 2 (2 jr + 1 )). The basis wave functions for a system of N nucleons in a single j-shell span an irrep {A} of U(2(2j + 1 )), where A is a regular partition of N. These basis wave functions can be labelled by quantum numbers relating to subgroups of U(2(2j + 1)). A subgroup of interest is the subgroup U ( 2 j + 1 ) x U(2)r of separate unitary transformations of space-spin and isospin wave functions. Now since nucleons are fermions, the Pauli principle restricts the iV-particle wave functions to be those that are antisymmetric with respect to particle permutations, and so, by Schur-Weyl duality, they must span the irrep { 1 7V} of U ( 2 (2 j + 1 )). Since the groups U(2j + 1) and U(2)x form a dual pair of subgroups within the irrep {lA } of the containing group U(2(2j -I- 1)), it follows from fermionic unitary-unitary duality that A-particle basis wave functions can be labelled by either their U (2 j + 1) quantum numbers or their complementary U( 2 )T quantum numbers. For example, consider the case N = 4. The full Hilbert space of physically acceptable 4-particle wave functions (i.e., that carries the irrep { l4} of U(2(2j + 1))) decomposes into subspaces that carry irreps of U (2 j + 1) x U(2)T as follows: {l4} ^ {l4} x {4} © {211} x {31} © {22} x {22} (3.31) A complete labelling scheme for basis states in each subspace of the type shown in the previous equation is discussed in Chapter 5. 3.4.2 L — S T coupling In the L — ST coupling scheme, spin and isospin wave functions for each particle are combined to produce a total spin-isospin wave function for the entire system. Also, spatial angular momentum wave functions for each particle are combined to produce a total spatial wave function for the system. Finally, the total spatial angular momentum wave function and the tota l spin-isospin wave function are combined to produce a total wave function for the system. Wigner [160] introduced the “supermultiplet group” U(4) as the group of all unitary transformations of the spin-isospin wave functions for a single nucleon. For light nuclei, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 3. U(m) x U(n) D u a l i t y 39 supermultiplet quantum numbers correspond to a physically relevant symmetry; for heavy nuclei, the strong spin-orbit coupling breaks this symmetry. In the supermultiplet scheme, a relevant group-subgroup chain for the labelling of spin-isospin wave functions in terms of their individual spin and isospin is U(4) 4 , U(2)s x U(2)t | U (l ) 5 x U(1)T (3.32) where the subscript S relates to spin and the subscript T relates to isospin. If there are p spatial wave functions, then total wave functions for the system belong to an irrep of the group U(4p), and part of the relevant subgroup chain is U(4p) D U(p) xU (4). Since tota l wave functions must be antisymmetric, the relevant irrep of U(4p) is {1 ^ }, where N is the number of nucleons, and once again we have duality present. Thus, the spatial and supermultiplet symmetries are in one-to-one correspondence. One set of physically relevant basis states is labelled by spatial angular momentum quantum numbers L and M l - There are a number of potentially relevant subgroup chains. For instance, if the single-particle states carry angular momenta labelled by I, then a relevant chain is u(PH n u(2/+i) (3-33) 1 States of N particles in an (l)N configuration can be labelled by making use of the subgroup chain U(2Z + 1)4 SO(2i + 1 ) 4 , SO(3) (3.34) An alternative, which is helpful for understanding the collective model in microscopic shell model terms is to make use of the chain U(p) 4 SU(3) ; SO(3) (3.35) where SU(3) is a symmetry group of the spherical harmonic oscillator. This option is useful when the spatial single-particle wave functions span a completemajor shell of the spherical harmonic oscillator. An essential element in the discussion of symmetry properties of iV-particle systems is the concept of symmetrized tensor product, which is discussed in the following chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Plethysm This chapter deals with a problem of fundamental importance in many-body physics: the construction of a m ulti-particle wave function (from single-particle wave functions) having a particular particle-permutation symmetry. Chapters 2 and 3 discuss situations where the single-particle wave functions span the fundamental irrep of a unitary group U(n); in these cases Schur-Weyl duality applies directly. In this chapter, we discuss such constructions when the single-particle wave functions span an arbitrary representation of a unitary group U(n). As we shall see, the solution to the problem is that the Hilbert space that carries an AT-fold tensor power of an arbitrary irrep of U(n) decomposes into subspaces, each of which carries a direct sum of irreps of U(n), and each subspace is characterized by a particular permutation symmetry A, where A is a regular partition of N. The direct sum of irreps carried by each such subspace is known as a plethysm , 1 or symmetrized power. It was noted in Chapter 2 that the process of decomposing a tensor product of unitary group irreps is related to the process of determining a branching rule from a unitary group to another unitary group, and that this relationship is mediated by Schur-Weyl duality. It follows from this that branching rules can be exploited to calculate plethysms. (Conversely, it is often convenient to use existing algorithms for plethysm to facilitate the calculation of branching rules.) Because branching rules can be determined via character theory by substituting one set of variables for another in the expression for a group character, plethysms can also be determined by such a substitutional procedure. As we shall see, this means that the plethysm of Schur functions is fundamental. 1In Littlewood’s original nomenclature, what we refer to as plethysm is known as outer plethysm, and the decomposition of a symmetrized power of a symmetric group irrep into permutation symmetry classes is known as inner plethysm. Since we do not discuss inner plethysm in this thesis, we refer to outer plethysm simply as plethysm. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4 . P l e t h y s m 41 The extension of the plethysm concept to other compact groups, and applications to the nuclear shell model, are also discussed in this chapter. The use of plethysm (and the Schuropera program that we have developed) in determining branching rules of interest in nuclear physics is illustrated in Chapter 5. In Chapter 6 the plethysm concept is extended to noncompact groups, with applications to the symplectic nuclear model. 4.1 Symmetric functions Once electronic computation became common, algorithms for computing the plethysm of Schur functions were devised based on manipulations of other symmetric functions. A discussion and evaluation of some of these algorithms is found in [24], This section contains a brief summary of some facts about symmetric functions needed later in the chapter. Besides Schur functions sx and power-sum functions px, which have already been defined, three other classes of symmetric polynomials are needed in this chapter. The first of these are the monomial symmetric functions m\, where A = (AiA 2 ... Ak), defined in terms of n variables as m^ = J 2 Xh Xi2 ' " Xik (4 1 ) where the sum includes all distinct terms obtained by all possible permutations of the n variables taken A: at a time. The homogeneous symmetric functions hx are defined in terms of the monomial symmetric functions as h\ = hXl h\ 2 hXk where hr = m A (4.2) A l-r and the sum includes all distinct monomials of degree r. The elementary symmetric functions er and the power-sum symmetric functions pr are defined as er = rapr) and pr = m(r) (4.3) where, as before, ex = eXl e X2 ■ ■ ■ eXk and similarly for the power-sum symmetric functions. By convention, if k > n then sx = mx = 0, and similarly if r > n then er — hr = 0. For simplicity, we refer to monomial symmetric functions as m-functions, power- sum symmetric functions as p-functions, and homogeneous symmetric functions as h- functions. The set Ar of all symmetric polynomials of degree r in n variables (together with the zero polynomial) is a vector space with respect to the operation of addition of functions. It can be shown 2 that each of the following five sets of symmetric functions forms a 2See Sagan [142, §4.3, §4.4] or Pulton [48, Chapter 6]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 42 basis for Ar : {m ^|A h r } , {hx\\ h r}, {eA|A h r}, A h r}, and {sa|A h r}. Since they all form bases for symmetric functions of degree r, it is possible to express each symmetric function in terms of any of the other types of symmetric functions. Including the operation of multiplication of functions makes the direct sum of the vector spaces Ar, where r = 0 ,l,2 ,..., into a polynomial ring. For three variables, the first few symmetric functions of each type are shown in Table 4.1 (by convention, m(o) = h0 = eo = Po = S(o) = 1)- In general, the elementary symmetric functions are identical to Schur functions whose Young diagrams have only one column (i.e., er = S(i«-)), and the homogeneous symmetric functions are identical to Schur functions whose Young diagrams have only one row (i.e., hr = S(r)). The reader may also observe instances of the general relations among the symmetric functions given earlier by comparing the expressions in Table 4.1. 4.2 The plethysm concept First we establish the link between plethysms and branching rules. Consider an irrep {A} of U(n), carried by an ra-dimensional Hilbert space V, where m > n. The matrices of this irrep are m by m unitary matrices, and as such, form a subset of the matrices of the fundamental irrep {1} of U(m). This is described by the corresponding branching relationship U(m) {1 } (4.4) For a concrete example, consider particles in the 2 hoo s-d shell of a spherical harmonic oscillator potential. There are six states in the level, and so the states transform among themselves w ith respect to U ( 6 ) transformations; in other words, these six states span the {1} irrep of U( 6 ). However, there is also a relevant U(3) symmetry, since the same six states span the {2} irrep of U(3). Thus, the appropriate branching relationship is U{6 ) {1 } I I (4.5) U(3) {2 } For a system of N particles in the s-d shell, the relevant representations are the tensor- power representations {1 }N of U( 6 ) and { 2 } w of U(3). If one wishes to identify a subspace of the Hilbert space for the system that corresponds to a particular particle permutation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4 . P l e t h y s m 43 ______Table 4.1: Selected symmetric functions of low degree in three variables m < i) Xi + £ 2 + £ 3 m ( 2) x \ + x% + x l i i ) X iX 2 + X iX $ + X2X3 nr* 3 [ /y»3 1 /y>3 m ( 3) X j \ d>2 I X ^ m ( 21) x\x2 + xjxs + X%Xi + x\x% + x \ x \ + x l x 2 J7i ( i n ) X1X2X3 e i X 1 + X2 + £ 3 e 2 X1X2 + X iX 3 + X2X3 e 3 X1X2X3 h i X i + £ 2 + £ 3 h 2 x \ + £2 + £3 + £ i£ 2 + ££ i 3 + X2X3 h$ x \ + x \ + £3 + £?£2 + £?£3 + £2£i + £2£3 + £ ^ i + £ ^2 + £ i£ 2£3 P i £ l + £2 + £3 /y»2 1 /Y"2 > /y>2 P2 X ^ X 2 X g 7*3 1 I /y>3 P3 Xj t X2 “ x^ 9(1) £ 1 T £ 2 T £ 3 9(2) x \ + £2 + £3 + £ i£ 2 + £i£3 + £2£3 S ( l l ) £ l£ 2 + £ i£ 3 +£ 2 £ 3 9( 3) x \ + £2 + £3 + £ j£ 2 +£ ? £ 3 + £ ^ £ i + £ 2 £ 3 + £3£i + £ ^2 + £i£2 £ 3 9(21) x \ x 2 + £ ? £ 3 + £ ^ £ 1 + £ 2 £ 3 + £ 3 £ i + £3 £ 2 + 2 a ; i £ 2 £ 3 9(11!) £ i £ 2 £ 3 symmetry, then one must determine the corresponding plethysm in the decomposition of the representation { 2 }w of U(3). Returning to the general situation described by equation (4.4), the iV-fold tensor power irrep {1}/V of U(m) branches to the iV-fold tensor power irrep {X}N of U(n). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 44 Applying Schur-Weyl duality to the tensor power {1} of U(m), we obtain = ( « ) ti\-N where Xe is the dimension of the irrep (/i) of SN. The permutation symmetry of a representation is not changed upon branching, and so each irrep {/i} in the decompo sition of the representation { 1 }^ of U(m) branches to a representation of U(n) that has permutation symmetry described by the irrep (//) of Sjy Thus, the A-fold tensor power representation {A }N of U(n) decomposes into representations characterized by the permutation symmetry {ji) of S n , termed plethysms, and symbolized 3 by {A} © {//}: U(m) {1} = x SM I 1 1 (4.7) U(n) {A} (A }* == E^-JV Xe [{A} © M ] Thus, given the basic branching relationship in the middle column of the following schematic diagram, each plethysm of the irrep {A} corresponds to the branching rule in the right-hand column: U(m) { 1 } I I (4.8) U(«) {A} {A } © {ft} Since the characters of the irreps involved in a branching relation can be determined by substitution of variables, one can determine plethysms by such substitutions. For example, consider the irrep {2} of U( 2 ). Since {2 } is 3-dimensional, we have the branching relation U(3) { 1 } (4.9) U(2) { 2 } A diagonal U(2) transformation acts on the 1-particle states that form a basis for the 3The notation for plethysm introduced by Littlewood [95] is {A} <8{n}. We introduce an alternative notation here to avoid confusion with tensor product, which is also denoted by 8 . The character of the representation {A}@ {fj,} is symbolized Sa@ sfl, and so plethysm can be considered to be an operation on characters just as well as an operation on representations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4 . P l e t h y s m 45 {1} irrep of U(2) as follows / X\ 0 1 «il°) ^ (4.10) 0 x2 4lo) and the irrep’s character is S(i)(x) = x\ + x2. The 2-particle states that form a basis for the {2} irrep of U(2) therefore transform as a M | 0 ) !->• xla\a\ |0 ) a|a||0 ) x\x2 ajc4 |0 ) (4.11) aJaJlO) - 7 rci <4<4|0> and the associated character is s2(x) = x\ + x\x2 + x\. A diagonal U(3) transformation acts on the 1-particle states that form a basis for the {1} irrep of U(3) as follows / Z\ 0 0 N ' d | 0 > ' ( Z\ c||0 ) ^ = (4.12) 0 2 2 0 4lo) 2 2 4|o) 0 0 23 , v 4lo> , ^ 2 3 4lo) j and so the irrep’s character is Si(z) = z \+ z2 + z3. Comparing (4.11) with (4.12), it follows that one can determine the characters of the representations of any branching associated with the basic branching (4.9) by making the substitutions Z\ xx, z2 X \X 2, and z3 x\ (4.13) For example, suppose that one wishes to determine the branching of the irrep {3 } of U(3) down to U(2); then one applies the substitutions (4.13) to the expression for S(3) in Table 4.1 to obtain s&(z) = z\ + z\z2 + z\z3 + zxz\ + 2:1 2 :2 2 3 + zizl + z\ + 2 :3 2 :3 + z2z\ + z\ (4.14) = x\ + x\x2 + x\x\ + x\x\ + x\x\ + x\x\ + x\xl + x\x\ -I- X\x\ + xl (4.15) The polynomial (4.15) can be expressed in terms of Schur functions as S(6 )(x) + S(4 2 )(x) (4.16) and therefore {2 } @ {3} = { 6 }® {42} (4.17) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4 . P l e t h y s m 46 In the same way one can determine the plethysms {2 } © {21 } = {51} ©{42} and {2} © { l 3} = {33} (4.18) and so the three-fold tensor power { 2 } 3 has been decomposed into its symmetry classes. The same technique can be used for all plethysms of unitary irreps, but computer pro grams based on this method would be quite inefficient. Practical algorithms for the computation of plethysms are discussed in Section 4.3. 4.2.1 Algebraic properties of plethysm Older methods for calculating plethysms rely on various algebraic properties of the plethysm operation, [165, p 52] some of which are summarized in this section. This section develops the principal algebraic properties of the plethysm of S-functions. The relevant properties are that plethysm is associative; is distributive on the right over or dinary multiplication, addition, and subtraction; and is not distributive on the left over multiplication, addition, and subtraction. Some other useful theorems on plethysm can be found in [165]. In the following equations, A, B, and C stand for Schur functions or linear combi- nations of Schur functions (for example, characters of orthogonal or unitary-symplectic groups). A© (B © C ) = (A@B)@C (4.19) A ® (BC) = (A © B)(A © C) (4.20) A@(B±C) = (A@ B)±(A@ C) (4.21) (A + B)®sx = ]T](Al©[sA/s<7 ])(£© Sa) (4.22) (j (A -B )© s x = £ (-l)H(^@[W^])GB@s,) (4.23) (j (AB) © sA = {A © [sA o sff]) (B © Sfj) (4.24) a where | / and o represent division and internal product 4 operations on Schur functions. It is sufficient to restrict the sums in the previous three equations to partitions a whose weights are less than or equal to the weight of A in equations (4.22) and (4.23), and equal to the weight of A in equation (4.24). The interpretation of the minus signs in equations (4.21) and (4.23) is discussed in footnote 11 of Section 4.4. 4The internal product of Schur functions is also frequently called inner product or inner multiplication; we avoid these terms as they can be confused with the scalar product defined on symmetric functions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4 . P l e t h y s m 47 The operation of division of Schur functions is related to the product of Schur func tions in the following way . 5 The result of s\/sa is a linear combination of Schur functions such that for each p, spsc = wpsx + ..., where the sum is finite and the co efficients wp are positive integers. That is, the irreps that correspond to the division sx/sa are those that, when combined by tensor product with the irrep {cr}, result in a representation that contains the irrep {A}. The internal product of Schur functions corresponds to the tensor product of the corresponding symmetric group irreps, via the characteristic map of Frobenius, mentioned in Chapter 2. That is, the result of the in ternal product sx o sa = Yin wTsT corresponds to the symmetric group tensor product (A)® (a) = More details about both operations can be found in [29] and [103]. Equations 4.22, 4.23, and 4.24 can be w ritten in the alternative forms: (.A + B)©sx = (4.25) li,v (A -B )® s x = (A @ s„) (B ® se) (4.26) fj,,v (.AB) @ sx = . £ {A © sp) (B © sv) (4.27) where the coefficients are taken from {p } ® {v} = Y2 \ T^{A} and the coefficients kpil are taken from the tensor product of symmetric group irreps (/i) ® (v) = Ylx^A^)- In the previous three equations, the summations include the cases {p } = {0} = 1 and {v} = {A}, and vice versa. The algebraic properties of plethysm can be understood by thinking of plethysm in terms of branching. For instance, equation (4.19) can be understood by studying the following schematic diagram: U(m) { 1 } C { 1 } C 4 1 1 (4.28) U(n) { i } BB@C 4- 1 i 1 1 1 m A A © B A © (B @ C) A © B (A ® B ) Equation (4.20) can be understood by studying the following diagram; a similar diagram 5Note that division of Schur functions is not exactly the inverse of multiplication of Schur functions. That is, it is not generally true that (sAS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 48 serves to explain equation (4.21). U(m ) { 1 } BC 4 4 4 4 (4.29) U(n) A A@ B A © C Equation (4.25) can be understood by studying the following two diagrams, and recalling from Chapter 2 the result, based on Schur-Weyl duality, for the branching rule U(m +n) 4 U(m) x U(n), namely {A} 4 E// „ Fj)„{/t} x { ^ } - 6 (For an explanation of this branching rule see the discussion of equation (2.16) in Chapter 2 .) U(m + n) U(m + n) {1} {A} 4 4 4 4 U(m) 4 4 4 {1} M 4 U(n) 4 4 4 4 { i } 4 4- 4 4 4 4 4 U(«)UM (A + B) (^4 + B) © {A } A A © M B U (m + n) {A} 4 4 U(m ) x U(n) E x M 4 4 U(g) xU(?) [A © {^}] x [B © M] 4 4 u 0s) E ^ * [A © M ] 6The type of situation illustrated in diagram (4.30) arises frequently in practice. For example, consider particles that can be in either of the f-p shell or the s-d shell in a spherical harmonic oscillator potential. The ten states of the f-p shell span the fundamental irrep {1} of U(10), the six states of the s-d shell span the fundamental irrep {1} of U(6), and the sixteen states taken together span the fundamental irrep {1} of U(16). Thus, the relevant group chains are U(16) 4 U(10) 4 U(3) and U(16) 4 U(6) 4 U(3) (that is, m = 10, n = 6, and q = 3.) This example will be revisited in Section 4.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4 . P l e t h y s m 49 the tensor product of the paired irreps of U( U(rnn) {A } I I (4-32) U(m) x U(n) k ^ { n } x { v } we recall7 the model of N particles with two attributes, such as spatial angular momentum and spin, and so all of the partitions A, /t, and u are partitions of N. By Schur-Weyl duality, the irreps {/ 1} and { v } that appear in the branching rule correspond to those irreps (/t) and (v) that, when combined, contain the irrep (A). But this is exactly what is meant by the coefficients k*u. To ju stify equation (4.26), one can solve for s,t in terms of s\ and sv in the branching rule U(m + n) j, U(m) x U(n), in the first two rows of diagram (4.31). The result is8 »<. = E ( - 1 )l‘'l r t A s e (4.33) A, v Now replace A by A — B in diagram (4.30) and make use of equation (4.33) to arrive at equation (4.26). Note that for arbitrary {A}, {A} © {0} = {0}, as can be verified by considering a schematic diagram sim ilar to the ones above. Similarly, for a partition w ith only one part k, then {0} © {&} = {0}, but if {A} has more than one part, then {0} © {A} = 0. This can be understood by thinking in terms of V (n) harmonic oscillator states; {0 } represents the ground state, which is non-degenerate. A partition with more than one part would represent an anti-symmetrization of at least two particles, but since there is only one state available, the result is necessarily null. Also note that {1} © {A} = {A} © {1} = {A}, both of which follow immediately from the fundamental correspondence (4.8) between plethysm and branching rules. 7See Section 1 of Chapter 3. 8See [96, p. 330]. One can also derive equation (4.33) in a straightforward way by explicitly writing out the branching rule for every A h TV, for the relevant value of TV, and then solving the resulting array of equations for each yu h TV of interest. See the appendix to this chapter for a derivation of equation (4.26) based on the same idea behind our new algorithm for plethysm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 50 4.3 A new algorithm for plethysm The substitution procedure for calculating plethysms of Schur functions given in the previous section can be used in principle as the basis of an algorithm for the automatic computation of all plethysms. However, as the degrees of the polynomials and their number of terms increase, the number of steps needed to effect such calculations grows at a rate that makes the method impractical even for automatic computation. The practical difficulties in calculating plethysms have stimulated a continuing search for algorithms that are both simple and efficient. Some milestones include a notable paper by Butler and King [14], in which they obtain recurrence relations for plethysms; the algorithm for plethysm of Chen, Garsia, and Remmel [24]; and Yang’s [166] method for evaluating the coefficient of a single Schur function in the expansion of a plethysm of two Schur functions. For introductory summaries of plethysm calculations with examples, see [29], [165], [98, pp 120-123, §XVI], and [97, Ch X and Appendix A]. Also, computer codes exist to evaluate Schur function plethysms; for example, SCHUR [146], and Schuropera [17]. Additional difficulties arise when one needs to calculate symmetrized tensor powers of a sum, difference or product of Schur functions, which we call compound Schur functions. Since the plethysm of a linear combination of Schur functions is not equal to the linear combination of the individual plethysms, the evaluation of such plethysms has up to now relied upon complicated manipulations of Schur functions, described later in this section. Physical situations where plethysms of compound Schur functions are required include the classification of states with particular permutation symmetries for a system of particles • in a multi-shell configuration in the nuclear shell model (described by irreps of unitary groups) • described by irreps of orthogonal or unitary-symplectic groups In this section we outline a new method [16] for calculating plethysms of Schur func tions that is conceptually simpler than previously developed methods. The method relies on a new algorithm for multiplying monomial symmetric functions, also developed in [16],9 and no other complicated rules intervene. Moreover the algorithm has been implemented in a MAPLE program [17], so it is easy to carry out plethym calculations interactively. A major strong point of the method is that plethysms of compound Schur functions are dealt with in exactly the same way as plethysms of single Schur functions. In this way the enormous complications of older methods, which can be appreciated by 9 The paper also presents a new way of calculating the change-of-basis coefficients between S-functions and m-functions (using Gel’fand patterns), and a new formula for m-functions in terms of p-functions. A new formula for the Littlewood-Richardson coefficients is also obtained. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 51 surveying some of the milestone papers mentioned previously, are avoided. The num ber of variables in which the Schur functions are expressed can be specified in advance, significantly simplifying the calculations in typical applications to many-body problems. Recall from Section 4.2 that the interpretation of the plethysm of S-functions in terms of branching relations for unitary irreps allows one to calculate the plethysm of S-functions by substituting the terms of one of the S-functions for the variables of the other. This idea can be extended to define the plethysm of any compatible symmetric functions, where compatible means that the number of terms of one of the symmetric functions is equal to the number of variables of the other. This generalized definition of plethysm is used in the derivation of a new algorithm for the plethysm of S-functions. The new method rests on the following key formula for the plethysm of an m-function w ith a p-function, " V © Pj = (4.34) where j./j, means that each part of n is m ultiplied by j ; that is, if /j, — (n i, ..., //*) then j.{i = ( jn u ..., j n k). For example, m ^ n ) © P 2 = ^ ( 8422)- It is also fundamental to the new method that one can transform easily among the various bases for symmetric functions. In particular, one needs to transform between S-functions and p-functions, and also between S-functions and m-functions. The former is a straightforward application of Schur-Weyl duality, and the transformation coefficients of the latter are known as Kostka coefficients, and there are known methods for obtaining them. The new method can be outlined as follows. By expanding the Schur functions on the left of the plethysm sign Q) (for example, see the left side of equation (4.22)) in terms of m-functions, and the S-function on the right side in terms of p-functions, the plethysm of a compound S-function (or of a single one) is then reduced to the plethysm of a series of m-functions w ith p-functions. But since the plethysm of an m-function w ith a p-function is still an m-function, evaluating the plethysm of S-functions (single or compound) only involves the multiplication of m-functions, which can be effected using a new algorithm developed in [16]. Let us illustrate the process with an example by considering the plethysm in U(3) ( S ( i ) + S(2) + 3 ( 3) ) © 3 ( 21) (4.35) This plethysm is relevant to the physical problem of finding the U(3) representations that can describe 3 particles placed with permutation symmetry (21) in any of the 1 hu>, 2hu), or 3 Huj valence shells of a spherical harmonic oscillator potential. We shall outline how the calculation is carried out using equation (4.22), and then using the new method, so that one may compare the two methods. (O f course both methods would be implemented by automatic computation in practice.) Previous to the development of the new method, one would apply equation (4.22) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 52 with, for example, A = S(i) + S(2) and B = S(3) to obtain ( 8 ( 1 ) + S(2) + 8(3)) © S(2i) = [(«(!) + S(2)) © «(21)] [«(3) © 8(0)] + + [(a(i) + S(2)) © S(0)] [s(3) © 8(21)] + [( 8 ( 1 ) + 8(2)) © 8(2)] [s(3) © S(1)] + + [(S(i) + 8(2)) © 8(1)] [s(3) © 8(2)] + [(s(i) + 8(2)) © 8(1*)] [s(3) © 8(1)] + + [ ( 8 (1) + 8 (2)) © 8(1)] [8 (3) © S (i2)] (4 .3 6 ) Now one applies equation (4.22) again to the first factor of each term in the preceding equation. Then one must determine each of the “simple” plethysms, which is done using one’s favourite algorithm. The result is a complicated combination of Schur functions, which one can expand by repeatedly applying the Littlewood-Richardson rule. The result is (8(1) + S(2) + S(3)) © S(21) = S(81) + 8(72) + S( 63) + 8 ( 621) + S(54) + 8(531) + 8(432) + + 8(8 ) + 2 S(7i ) + 3 S(62) + 5 (612) + 3 S(53) + 2 S(52l) + S(42) + 2 S(4 3 i ) + S (4 2 2) + S (322) + + 2S(7) + 4S(6i) + 58(52) + 2 S(512) + 48(43) + 3S(42l) + 2 8 (324) + S( 322) + + 2 8 ( 6 ) + 5 8 (51) + 5 S(42) + 2 S(4i 2) + 2 S (32) + 3 S(32l ) + + 2S(5) + 48(41) + 38(32) + 2S(3 i2) + S(22i ) + + 8(4) + 2S(3i) + S(22) + S(2i2) + + S(2 i) (4.37) The general result in the previous equation is also valid for U(3) since there are no partitions that have more than 3 parts. Compare the previous calculation w ith the following one, which is based on the new method. First express S( 2 i) in terms of p-functions, and express S(q + S(2) + S(3) in terms of m-functions. Since we are working in U(3), we restrict the m-function labels to three parts or less. ( « ( i ) + «(J) + 3 (3 )) ® S ( 2 1 ) = (n»(i) + n > (2 ) + !»()■) + ">(») +i » ( 2 i ) + ^ (pp>) - Pry,) (4.38) Using equation (4.34) and the property that plethysm is distributive on the right (equa tions (4.20) and (4.21)) one obtains 1 3 (8(1) + 5(2) + 8(3)) © 8(21) — g (m (l) + m ( 2) + m (l2) + m (3) + ” 1(21) + m^S)) — - ^ (m(3 ) + m(6 ) + m (32 ) + m (9) + m ( 6 3 ) + m (33 )) (4.39) Now one expands the expression involving m-functions on the right side of equation (4.39), collects like terms in the resulting sum, and then converts all the m-functions to S-functions. The result is the same as before. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 53 I have recently become aware 10 of an optics problem that illustrates well the utility of both the plethysm concept and the Schuropera program. The experimental situation involves photons whose states are linear combinations of two basis states, | H) and If7), where the former represents horizontal polarization and the latter represents vertical polarization. The photons are all in the same spatial mode, and therefore have no other relevant degrees of freedom. The Hilbert space H for an individual photon is spanned by the states \H) and|V), and H carries the {1} irrep (that is, the spin- ~ irrep) of SU(2). In order to perform relevant calculations on an TV-photon system, one needs to know the transformation properties of the linearly independent operators acting on the iV-fold tensor power space IP that are symmetric with respect to particle permutation. Such operators can be constructed as iV-fold tensor powers of operators that act on H. Each of the latter operators has the same transformation properties as the tensor product of two photons; that is, it transform as the {1} ({1 } <8 ) {1 }) ® { N j (4.40) By the Littlewood-Richardson rule, {1} © {1} = { 2 } © {11}, but, for SU(2), {11} must be modified to {0 }. The plethysm problem in equation (4.40) becomes ({2}© {0}) ® { N } (4.41) This is easily solved with the help of Schuropera, with the result ({2} © {0}) ® {TV} = 0 k [{2(iV - 2 k + 2)}] © [{2(JV - 2 k + 1)}] (4.42) where SU( 2 ) labels are used in the previous equation, and the direct sum includes all positive labels. A potentially tedious problem was solved directly when expressed in terms of plethysm and with the aid of Schuropera. Finally, many branching rules can be recast in terms of plethysms, so that an available algorithm for calculating plethysms can be used to determine branching rules as well. For example, consider the basic branching U(n) {1 } 4. 4 (4-«) U (n -1 ) { l} + {0} 10I thank Peter Turner for communicating the problem to me and I thank Peter Turner and Rob Adamson for explaining the physical context. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 54 Then the branching rule for an arbitrary irrep {A} of U(n) is U(n) {A} 4- 4- (4-44) U (n-l) ({l} + {0» ®{A} The calculation of such plethysms are straightforward in principle: express the sum of S-functions to the left of the plethysm symbol in terms of m-functions, express the S- function to the right of the plethysm symbol in terms of p-functions, and then apply the new algorithm. 4.4 Plethysm for other compact groups Plethysm is a quite general concept and applies to the decomposition of any tensor power of any irrep of any group. However, plethysm was first defined in the context of Schur functions by Littlew ood [95], who developed a number of useful techniques for its calculation [97]. The plethysm of Schur functions applies directly to the decomposition of iV-fold tensor powers of irreps of unitary groups. It is well known [165] that the characters of the unitary irreps of all the other Lie groups of interest in this thesis are expressible in terms of Schur functions, and vice versa. To calculate a plethysm for any such group G, then, one expands the characters of G in terms of Schur functions, determines the plethysm of the Schur functions, and then expresses the resulting series of Schur functions back in terms of the characters of G. This procedure is straightforward in principle for compact groups, although the calculations may be difficult in practice. For non-compact groups, where the characters are infinite series of Schur functions, the difficulties are considerably greater. Methods for evaluating plethysms of, for example, the fundamental irrep of non-compact symplectic groups have been discussed in [15, 162, 111, 55, 81]. These methods enable one, in principle, to obtain the fu ll infinite-series expansion of such plethysms, though in practice one has to truncate them at a prescribed cutoff. In any case, the plethysm of Schur functions is of fundamental importance for the calculation of plethysms for all groups of interest in this thesis. The following example illustrates the general procedure for compact groups. Consider the SO(3) plethysm [ 2 ] © [2 ]. The first step is to convert the SO(3) irrep [ 2 ] into SU(3) irreps. In light of the branchings SU(3) { 2 }{ 0 } (4.45) I 1 SO(3) [2 ] © [0 ] [0 ] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 55 one writes that {2} — {0} | [2]. This is a convenient notational convention that is justified by its validity at the character level. That is, the character of the SO(3) irrep [2] is identical to the polynomial s 2 — s0, when the latter is restricted to SO(3).u The next step is to determine the plethysm ({2} — {0}) © {2}. Using equation (4.23), «2}-{0})® {2} = ({2}®{2/2})({0}®{l2})-({2}@ {2/l})({0}@ {l}) + + ({2}®{2/0})({0}©{0}) (4.46) = {2}®{2}-{2}®{l} (4.47) = {4} + {22} - {2} (4.48) The next step is to reduce the SU(3) irreps back down to SO(3) irreps, using the branching rule. The result is ([4] + [2] + [0]) + ([22] + [2] + [0]) - ([2] + [0]) (4.49) The label [22] is nonstandard for SO(3) and must be modified . 12 Applying the modifi cation rule for SO(3) indicates that the label [ 2 2 ] should be discarded. Thus, the final result is [2] © [2] - [4] ® [2] © [0] (4.50) A similar calculation shows that [2] © [I2] = [3] © [1] (4.51) and thus the two-fold tensor power [ 2 ] 4.5 Applications of plethysm in the nuclear shell model Situations where plethysm calculations arise in the nuclear shell model involve, for ex ample, 11The six states of the s-d shell (that span the {2} irrep of U(3)) consist of five states that span the [2] irrep of SO(3) and one state that carries the [0] irrep of SO(3). One can thus also interpret {2} — {0} j. [2] to mean that if one removes the single state that carries the [0] irrep of SO(3) from the six states that span the {2} irrep of U(3), one is left with five states that span the [2] irrep of SO(3). Such an interpretation in terms of removal of states is sometimes useful, but care must be taken to avoid misconceptions. For instance, the single removed state in this case does not carry the {0} irrep of XJ(3). 12 One of the advantages of using character theory (Schur functions) to determine branching rules is that the results obtained are typically valid for groups of nearly all ranks. For example, the SU(n) I SO(n) branching rule {22} 4 [22] ® [2] © [0] is valid for n > 4. However, for n = 2 or n = 3, the label [22] is nonstandard. There are established procedures, called modification rules, for determining whether such nonstandard labels are to be discarded or retained, and how to convert retained labels to standard ones. For a comprehensive discussion of modification rules see [10, §3]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4 . P l e t h y s m 56 • m ultiple particles in a valence shell • multiple particles that can be in either of a number of different shells (i.e., multi shell configurations) Several representative examples of the use of plethysm in nuclear physics applications are discussed in this section. Recent papers discussing such applications include [18], [19], and [20]. Other applications of plethysm are discussed in [164]. A t the beginning of Section 4.2, the situation of several particles in the s-d shell of a spherical harmonic oscillator potential was mentioned. The spatial wave function for a single particle in the s-d shell corresponds to one of the states in the space V that carries the irrep {2} of U(3). For a system of N such particles, the spatial wave function corresponds to one of the states in the space VN that carries the tensor-power representation { 2 }^ of U(3). In practice, the spatial wave function is combined with wave functions for other attributes of the system, in such a way that the total wave function is either symmetric with respect to particle permutations (for bosons) or antisymmetric (for fermions). Thus, one needs to know the particle permutation symmetry of the spatial wave function, so that it may be combined with other factor wave functions with appropriate symmetries. The subspace of VN that carries wave functions with particle- permutation symmetry A is specified by the plethysm { 2 } © {A}. The following table, listing the plethysms for the first few values of N , was generated using the program Schuropera. { 2 } © { 1 } = { 2 } { 2 } © { 2 } = {4 } 0 {22} { 2 } © { 1 1 } = {31} {2} © {3} = { 6 } © {42} 0 {222} { 2 } © { 2 1 } = {51} ©{42} ©{321} { 2 } © { 1 1 1 } = {411} ©{33} {2} © {4} = { 8 } © {62} © {44} © {422} {2} © {31} = {71} © {62} © {53} © {521} 0 {431} © {422} { 2 } © { 2 2 } = {62} © {521} © {44} © {422} { 2 } © { 2 1 1 } = {611} © {53} © {521} © {431} © {332} The irreps in (4.52) belong to U(3), and so four particles placed in the s-d shell cannot have the permutation symmetry ( 1 1 1 1 ), since their wave function would necessarily be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 57 identically zero. Thus, the entry { 2 } © {1111} does not appear in the table. The table can be extended indefinitely, both for the s-d shell and for any other shell. Now consider two particles that can be in any of the states of either the 2 tuv s-d shell or the 3 tko f-p shell. The relevant schematic diagram is a t—i o f( 1 0 + 6 ) + { 1 } {A} I i 1 1 U(10) I { 1 }M 4 U(6 ) I 4 { 1 } M 4 I 4 I U(3) U(3) ({3} © {2}) ({3} © {2 })® {A} {3} {3} © M { 2 } { 2 } ® - (4.53) Now if the wave function for the two particles is required to be symmetric w ith respect to particle permutation (i.e., A = (2)), then the possible wave functions are those found in the plethysm ({3} ffi{2}) © {2 } (4.54) Using the Schuropera program, the result is ({3} ® {2}) © {2} = { 6 } ® {42} © {5} © {41} © {32} © {4} © {22} (4.55) Similarly, if the wave function must be antisymmetric, the result is ({3} © {2}) © {11} = {51} © {33} © {5} © {41} © {32} © {31} (4.56) The states in the subspaces specified by these plethysms can be classified according to their angular momentum by making use of the branching rule for U(3) {, SO(3). In the supermultiplet classification of states discussed in Chapter 3, basis states are labelled by making use of the subgroup chain U(4) D U(2)s x U(2)r (4.57) where the S subscript indicates spin and the T subscript indicates isospin. As was discussed in Chapter 3, the fundamental irrep of U(4) decomposes as U(4) {1 } I ; (4-58) U(2)s x U(2 )t { 1 } x { 1 } Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 58 Thus, an arbitrary irrep {A } of U(4) decomposes as U(4) {A} I (4.59) U(2)s x U(2)t ({1 } x {!}) © {A} Suppose that one wishes to combine a spin-isospin wave function w ith a spatial wave function with particle permutation symmetry ( 2 1 1 ) to produce a total wave function that is anti-symmetric with respect to particle permutation. Then one must choose the complementary symmetry A = (31), and the possible spin-isospin wave functions are those that span the representation ({1} x {!}) © {31} (4.60) The plethysm in the previous equation can be computed with the aid of equation (4.24), as follows : 13 (AB) © sx (4.61) a «1}X{1}) ©{31} = £ ( { i}®[{3w }])x ({i} © M ) (4.62) a (4.63) a {31 o 4} x {4} ©{31 o31} x {31} ©{31 o22} x {22} (4.64) {31} x {4} © [{4} © {31} © {22}] x {31} ©{31} x {22} (4.65) Most of the examples of plethysm calculations discussed in this chapter are taken from nuclear physics contexts, but sim ilar problems arise in atomic physics [149], particle physics [35], and other fields, and the same methods can be used to solve them. For applications of plethysm to measures of entanglement in quantum computing, see [31] and [32]; for an application to knot theory see [ 1 1 0 ]; for applications to the analysis of the Riemann tensor, curvature tensors, and tensor analysis in general, see [35, 46, 47, 43, 44], Further applications of plethysm to jV-body systems include [15, 111, 37, 55, 56, 59], and are further discussed in Chapter 6 . 13The notation {A o /i} is used to stand fors \ o sM. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4. P l e t h y s m 59 4.6 Appendix: A derivation of equation (4.26) using character theory Equation (4.26) can be derived by using the same idea upon which the new algorithm for plethysm is based. Begin w ith (-B) (4.66) and express B in terms of m-functions and s„ in terms of p-functions to obtain (■- b ) © Sv = ^ Y ^ gpx ppp j (4-67) where v h N , p specifies a conjugacy class (with gp elements) of the symmetric group Sjv , and x P is the value of the character of the representation (v) of Sjv for the conjugacy class p. Since plethysm is distributive on the right, (-B) © su = j^ J T , 9 pX up -'Y^otprrip J @ p p (4.68) p / (U p , (4.69) TV! J © p / \ j = 1 -'Y^otp.mp @pP: (4.70) = j= l jj. ^Ew;(-i)‘II (4.71) p j =i Now, the characters of conjugate irreps of Sjv are related by xs = ( - i r Mx, (4.72) where (—l)7rW is the parity of the permutations in the conjugacy class p. The parity of a permutation is related to the number of parts k of p through the relation 7 r(p) = N — k (4.73) Thus (4.74) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 4 . P l e t h y s m 60 Substituting equation (4.74) into equation (4.71) one obtains = pE s,*;(-i)'n Y l a Pmp ® ppj (4.75) P 3=1 ( - i f [B © sp] (4.76) since \v\ is defined to be N. Replacing B by — B in equation (4.25) and making use of equation (4.76) one arrives at equation (4.26), which completes the derivation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 USp(2fi) x USp(2u;) D uality The pairing interaction in nuclear physics describes the tendency of nucleons to couple into zero-angular-momentum pairs. Operators that create or annihilate pairs of fermions coupled to zero angular momentum are infinitesimal generators of unitary-symplectic groups, and so the latter are important in descriptions of nuclear structure involving pairing interactions. In shell model descriptions of nuclei for which j j coupling is appro priate, unitary-symplectic groups are also useful as intermediate subgroups in the chains used in classifying basis states. For nucleons in a single j-shell, where j is half of an odd integer, a relevant unitary- symplectic group is USp(2fi!), where = j + Helmers [63] discovered that associated with this group, and providing a complementary description of the physical situation, is another unitary-symplectic group USp(2w), where u) is related to the isospin r by oj = 2t + 1. Helmers’ discovery amounts to the duality of the groups USp(2fl) and USp(2o;) in their actions on a H ilbert space spanned by all A/’-fermion states, where N = 0,l,2,...,2Qw. 5.1 Unitary-symplectic operators and fermion-pair algebras 5.1.1 The use of quasispin and seniority in pairing problems Pairing interactions act to couple pairs of particles. To describe such interactions in the context of nuclear physics, one needs groups that have infinitesimal generators that 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2f2) x USp(2w) D u a l i t y 62 create and annihilate pairs of fermions, of the form ajaj and apaq. By determining their commutation relations, one can see that such operators close on a Lie algebra, provided that one includes operators of the form o)paq — \5pql. If the subscripts p and q can take on any of In possible values, then the algebra spanned by the operators (5.1) is the Lie algebra of the group 0(4n). Consider nucleons of a single type (either neutrons or protons — there is no need to explicitly consider isospin until the next section) in a single j-shell, and interpret the operators 'i)., as creating such a nucleon w ith angular momentum j and 2 -projection of angular momentum m. In the simplest discussions of pairing, one considers that the pairing interaction couples particles to zero angular momentum, and so the relevant o(4n) operators are (5.2) m > 0 which creates a pair of fermions coupled to zero angular momentum, and (5.3) which annihilates such a pair . 1 A direct calculation shows that (5.4) where (5.5) and since [ with the algebra they span is known as a quasispin group, symbolized SU( 2 )5. One notes that this coupling of particles in pairs, all of angular momentum zero, is possible only if the total number of creation operators is even; that is, only if the number of possible values of m is even. Since the number of possible values of m is 2j + 1 , this kind of construction is possible provided that j is half of an odd integer. This explains why p and q were chosen to range over an even number 2 n of possible indices in the previous paragraph. Multi-particle states can be labelled by their quasispin quantum numbers, by analogy with ordinary angular momentum; that is, the total quasispin S and the eigenvalue of 1The factor (—1)J m is included in equations (5.2) and (5.3) for consistency with the Condon-Shortley convention for coupling angular momenta. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 5. USp(2Q) x USp(2u;)D u a l i t y 63 So- From equation (5.5), one sees that the eigenvalue of So and the number of particles N for a state are related by So — N — ^(2j + 1) (5.6) Since the eigenvalue of So varies from a minimum of —S' to a maximum of S for states spanning the irrep labelled by S, the number of particles in a state of this irrep varies from a minimum of j + \ — S, which is defined as the seniority v, to a maximum of j + | + S. Thus, states can be labelled by v and N as an equivalent alternative to the labelling in terms of S and S0. For each quasispin irrep, the operator S- annihilates a state with a minimum number of particles. Thus, the physical interpretation of the seniority v of a state is the number of particles not coupled to angular momentum zero. The remaining basis states of the irrep have the same seniority, but differ in the number of pairs of particles coupled to zero angular momentum. Since the energy spectrum depends only on seniority, states w ith the same seniority have the same energy. The use of seniority in labelling states leads to the question, is there a group for which the infinitesimal generators are associated w ith seniority, such that the group transformations leave all quasispin quantum numbers unchanged? Such operators would have to commute w ith all of the quasispin operators, since each basis state in a quasispin irrep has the same seniority. A straightforward, if tedious, calculation shows that the set of all operators within o(4n) that commute with all the quasispin operators is spanned by operators of the form Cki = afkai + ( - l) ft+i (5.7) By determining the commutation relations satisfied by the operators (5.7), one can iden tify them as infinitesimal generators of the rank-n unitary-symplectic group USp(2n), also known as a compact symplectic group. The action of the Cm operators on basis states preserves the number of particles; because of the form of the Cm operators, one can see that USp(2n) is a subgroup of the U ( 2 n) group that is spanned by operators of the form a\ai — \ 8m^- Since the Cm operators commute with the quasispin operators, the former do not change the number of pairs of particles coupled to zero angular momentum, and therefore do not change the seniority of a state. Thus, irreps of USp(2n) can be specified by seniority, and different states within an irrep have the same number of particles, although the unpaired particles are in different states. The quasispin irreps can also be specified by seniority, but different states within an irrep have different numbers of particles. The abstract definition of a unitary-symplectic group is as a subset of unitary group operators acting on a 2n-dimensional vector space W that keep invariant a certain anti symmetric bilinear product. In the specific realization of unitary-symplectic group oper ators discussed so far, 2 n vectors that span W can be taken to be a)JO). The fact that the Cm operators commute with the quasispin operators makes clear that the antisymmetric Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2S7) x USp(2w) D u a l i t y 64 product in this case is X)m>o(— maL a-m • The properties of fermion operators ensure that the product is antisymmetric. The Lie algebra of the lowest-rank unitary-symplectic group USp(2) is isomorphic to su(2), and so the quasispin group SU(2)s is also a unitary-symplectic group, where the anti-symmetric bilinear product that remains invariant is defined by the expression for the Cki operators in equation (5.7). Thus, there are two different types of realizations of unitary-symplectic Lie algebras within a fermion-pair algebra. The complementarity between quasispin and seniority quantum numbers described in this section is generalized to the complementarity of the quantum numbers of a dual pair of unitary-symplectic groups by a theorem of Helmers, discussed in the next section. As for other compact groups, the unitary irreps of unitary-symplectic groups are finite dimensional, and each can be uniquely specified by the weight of a highest-weight state. Irreps for USp( 2 n) have highest weights that are regular partitions w ith n parts or fewer. Equivalently, irreps for USp(2n) can be specified by Young diagrams with n rows or fewer. The use of unitary-symplectic groups in atomic and nuclear spectroscopy was pio neered by Racah [120], and developed by a number of researchers, including Flowers [45], Kerman [75], Helmers [63], and Lawson and Macfarlane [8 8 ]. The discovery of group du ality among unitary-symplectic groups in general is due to Helmers [63], and was further explored by Moshinsky and Quesne [107]. Overviews of pairing and the use of unitary- symplectic groups in describing pairing can be found in [61, §lc], [114, §11], [60, §10, §11], [93, §5], and [94, §6 , §10]. For recent reviews discussing pairing in nuclear systems, see [33] and [155, §3]. 5.1.2 Generalizations of quasispin and seniority to groups of higher rank The preceding discussion is relevant for nucleons of a single type; to consider nucleons of both types, one needs operators that create and annihilate each type of nucleon. A convenient way to do this is to introduce a second index for each operator to distinguish which type of nucleon is created or annihilated. This second index thus represents isospin, with value + | for protons and —~ for neutrons. More complex applications of pairing involve couplings of particles for which the total isospin is greater than | , and so it is of value to consider arbitrary isospins in the general formulation of USp-USp duality that we now turn to. As before, the unitary-symplectic groups of interest have infinitesimal generators that lie w ith in fermion-pair algebras. Using fermion operators that have two indices, the fermion-pair algebra o(4fIw) is spanned by the following 2Quj(4Quj — 1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2D) x USp(2cu)D u a l i t y 65 operators, where the creation and annihilation operators are fermion operators: alpaiq ~ \b kidpqI (2 Dw) 2 operators a\pa\ Qu;(2Qu; — 1) operators (5-8) akpaiq Dw(2 Dw — 1 ) operators where k and I are in the range 1 , 2 ,..., 2 D and p and q are in the range 1 , 2 ,..., u>. There are 22Uw basis states for 2 Dw fermions in a single j-shell, vacuum state |0 ) one-particle states | 0 ) two-particle states “L4j°> (5-9) 2Siw-particle states J O ) and the collection of all of the 22nw states span a Hilbert space H that carries what is called the fundamental spin representation of the fermion-pair algebra o(4Dw). As noticed and proved by Helmers [63], the subgroups USp(2D) and USp(2w) of 0 (4 Dw) aredual with respect to their actions on BL Thus, we let Q = j + |, and uj — 2 t + 1, where t is the isospin. The operator a \ p creates a particle w ith angular momentum k and isospin p. Then for the operators listed in (5.8), k and I are in the range — j + — j , and p and q are in the range —r, — r + 1, ..., t — 1, r. The usp(2a>) subalgebra of o(40w) is spanned by the u)(2oo + 1) operators A« = ^ 2 ( aL a-mt + + 1) operators (5.10) m > 0 B sf, — ^ ] ( l) (o_TOSttOT( T u_m(flTOS) 2 w(w T 1) operators (5.11) m > 0 c st = ^ 2 ( alnSamt - ^5stl J w2 operators (5.12) m ^ ' The subscripts s and t range from —r to r, and m ranges from —j to j. Note that A st = Ats and B st = B ts■ Also note that the Cst operators span a u(ui) subalgebra of usp( 2 aj). The A st operators are all raising operators and the B st operators are all lowering operators. Among the Cst operators are raising operators (for s > t), lowering operators (for s < t), and weight operators (for s = t). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 5. USp(2!2) x USp(2w)D u a l i t y 66 The operators in equations (5.10-5.12) satisfy commutation relations characteristic of a unitary-symplectic algebra: [Cst, Cuv\ = ^lll Csv 3SV Cut, (5.13) [Cstl -^uw] = fitU-A-sv T &tvAsu (5.14) [C « u B uv ] 5suB tv SsvB tu (5.15) \B g tj j — $tu,B'vs $ti)Cus ^ s u ^ v t ^svC^ut (5.16) The usp(2H) subalgebra of o(4flw) is spanned by operators of the form T Ckl = (4rair + (- l ) k+l a iJra_*r) (5.17) T — — T where k and I range over — j, —j + 1 ,..., j — l , j and r ranges over —r, —t + 1 , ... , r —l,r. Because of the identity C-t,-k = ( - l) k+lCkl (5.18) only 1 2 (2 1 1 + 1 ) of the Cki operators are independent, and these span usp( 2 1 2 ). They can be chosen as in Table 5.1. It is straightforward to show that the operators in Ta- Table 5.1: Infinitesimal generators of USp(2Q) operators conditions number of operators k > 0 , I < 0 , and \k\ > |2 | (raising) •4|k||J| = *^1*11*1 ~ (“ l ) l B\k\\i\ = B \m = ( - l ) |fc|+2 Cw k < 0 , I > 0 , and \k\ > 2 | (lowering) |H(H + 1) Chi k > 0 , I > 0 , and k > I (raising) §0 ( 0 - 1) Cki k > 0 , I > 0 , and k < I (lowering) §0 (0 - 1 ) Ckk k > 0 (weight) o ble 5.1 satisfy commutation relations analogous to those in equations (5.13-5.16) that characterize unitary-symplectic operators. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2f2) x USp(2w) D u a l i t y 67 5.2 The USp(2f2) x USp(2w)duality theorem The USp(2f2) xUSp(2w) duality theorem can be stated in terms of the following branching rule, where A denotes the fundamental spin representation of 0(4f2w ) : 2 0(4Qu;) | USp(2Q) x USp(2w) A + ]T(A) x (A*) (5.19) The sum extends over partitions A_with at most Q, parts, each part is at most w, and A and A* are related by the rule (A*) = (w — An, w — A q-i, ..., w — A2, cu — Ai). In the application of this rule, if A has K parts, where K < then we set A^+i = • • • = An = 0. Helmers has expressed the relation between A and A* in terms of Young diagrams. To determine A* given A, begin with a rectangular array of boxes having Q, rows and u; columns. Place the Young diagram for A in the upper left corner of the array, and then remove the boxes corresponding to A. Reflect the remaining boxes in a horizontal line, rotate the figure counterclockwise 90°, and the resulting figure is the Young diagram for A*. For example, consider the case for USp( 6 ) x USp(4); that is, £7 = 3 and to = 2. Then the Young diagram for the partition A = (11) is placed as indicated by the X symbols in the following 3 x 2 array X X 1 Removing the indicated boxes, and reflecting and rotating as prescribed, one obtains the Young diagram for the partition A* = (31).3 Helmers’ complementarity theorem makes clear that within the Hilbert space HI that carries the fundamental spin representation of 0(4Qw), the resulting irreps of USp(2f2) and USp( 2 w) are in one-to-one correspondence, so that the decomposition of H does indeed result in a duality correspondence. In the example of USp( 6 ) x USp(4) acting on the Hilbert space of the spin representation of 0(24), the decomposition is A 4, (222) x (0) + (221) x ( 1 ) + (22) x ( 1 1 ) + (211) x (2) + (21) x (21) + +(111) x (3) + (2) x (22) + (11) x (31) + (1) x (32) + (0) x (33) (5.21) For a proof of the USp(2f2) x USp(2w) duality theorem see [63]. Another useful expression of this duality relationship in terms of Schur functions is found in Cummins [29, rule 44]. 2By convention, irreps of unitary-symplectic groups are denoted by partitions enclosed by angle brackets. 3 Helmers called the two Young diagrams that result from this construction complementary, which led to the usage of the term complementary groups for dual groups. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2fl) x USp(2w) D u a l i t y 68 5.3 Example: USp(4) x USp(4)duality on the space of the 0(16) spin representation In this section we illustrate the duality theorem in a low-dimensional case for which Q = u = 2. This example is relevant for a shell-model description of protons and neutrons (i.e. isospin r = |) in a single shell for which the angular momentum is j = |. The arena for this duality is the Hilbert space H that carries the fundamental spin representation of 0(16), which has dimension 2 8 = 256. Among the 256 basis states that span H are the vacuum state| 0 ); 8 one-particle states of the form aj.p| 0 ), where k — ± 1 , ± § and p = ± |; 28 two-particle states of the form a\pa\q\fy\ 56 three-particle states; 70 four- particle states; 56 five-particle states; 28 six-particle states; 8 seven-particle states; and one eight-particle state: ^ a 3 q\ i ci\ _ i i iO^_ i _ i cJ_ 3 1 3 _ 1 10) (5.22) 22222222 22 22 22 22 To distinguish the two USp(4) groups, we call one of them the “generalized quasispin” group (USp(4),g), and the other the “generalized seniority” group (USp(4)y), since the form of the USp(4)g operators is analogous to the quasispin operators and the form of the USp(4)v operators is analogous to the quasispin invariants (associated with seniority) discussed earlier in this chapter. From equations (5.10-5.12), 10 operators that span the Lie algebra usp(4)s are A *t = 5 Z (5-23) m > 0 B sf — y ( 1) (o,—mso,mi + Q,—miQ,ms) (5.24) m > 0 Cst — ^ ^ ( alnsamt ~ 2 ^st / (5.25) m ' ' where m = — |, — |, |, | and s, t = — Evidently each A st operator creates a pair of nucleons w ith total angular momentum coupled to zero, and w ithisospin characterized by s and t. Similarly, each B st operator annihilates such a pair. The Cst operators span a u(2) subalgebra of usp(4)s and do not change the number of particles. The actions of the usp(4)s weight operators on any of the basis states can be deter mined using the following commutation relations: [C.5,5, — 3sqalq i [C.S.S] apqauv\ = T ^sv) apyfJ'uv (5.26) and so on. Thus, the first component of the weight of a state with respect to the usp(4)s algebra is obtained by subtracting 2 from the number of occurrences of 5 in second Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2fl) x USp(2o;)D u a l i t y 69 indices of a) factors. The second component is likewise obtained by subtracting 2 from the number of occurrences of — \ in second indices of a) factors. We call this weight the S-weight of a state. The actions of the usp(4)s raising operators can be summarized by the following diagram, which shows the states of the fundamental four-dimensional (1) irrep. To simplify the notation we have used the subscript + to stand for - and the subscript — to stand for — |. Figure 5.1: Weight diagram for the (1) irrep of usp(4)s + - (0, 1) (1, 0) ++ • + - ■ + - (- 1,0) (0,-1) The states in Table 5.2 span one of the sixteen copies of this irrep that are found in H. Table 5.2: A copy of the (1) irrep of usp(4)s found in H State S-weight A-|—O3 j O3 1 o 1 110) (1 , 0 ) 2 2 2 2 2 2 A-i—O31 a 1 10 1110) (0 , 1 ) 22 2 2 22 O 3 1 ^3 1 & 1 1 |0) (0 ,- 1 ) 22 2 2 22 ati at 1 at 1 10) (- 1 , 0 ) 22 2 2 22 From Table 5.1, 10 operators that span usp(4)y are as follows; raising operators: *4 .3 3 = Cz 3 , *43i = —Cs_i, A n = —Ci_i, and Csi; lowering operators: Bss = 22 2 2 22 2 2 22 , 2 2 , . 2i2 n i n 2 2 4 C 3 3 , # 3 1 = C 3 1 , B ii = —C_i i , and C1 3 ; and weight operators: C 3 3 and C1 1 . A 2222 2222 22 22 22 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2f2) x USp(2w) D u a l i t y 70 straightforward calculation shows that [Ckh Cpq\ = filpCkq — hqCpl + (~ 1)P+9 (Si-gCk,-p — 5k-pC-ql) (5-27) Using this relationship it can be shown directly that the operators that span usp(4)y satisfy commutation relations of the same form as those satisfied by the operators that span usp(4)s. The actions of the usp(4)y weight operators on any of the basis states in H can be determined using the following commutation relations: \Ckkt apq\ = {$kp ~~ $k,-p) alq: \f-kk, a \>q a ^u,v\ = {^kp ~ $k-p + ^ku ~ &k,-u) apqahv (5.28) and so on. Thus, the first component of the weight of a state with respect to the usp(4)y algebra is obtained by subtracting the number of occurrences of — § from the number of occurrences of | in the first indices of factors. The second component is obtained by subtracting the number of occurrences of — \ from the number of occurrences of \ in the first indices of a) factors. We call this weight the V-weight of a state. The actions of the usp(4)y raising operators can be summarized by the diagram in Figure 5.2, which shows the states of the fundamental four-dimensional (1) irrep of usp(4)y. Figure 5.2: Weight diagram for the (1) irrep of usp(4)y (0, 1) The states in Table 5.3 span one of the sixteen copies of the irrep (1) of usp(4)y that are found in EL Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2fl) x USp(2u;)D u a l i t y 71 Table 5.3: A copy of the (1) irrep of usp(4)y found in H State V-weight a\ 110> (1 . 0 ) 2 2 a t ! |0 ) (0 , 1 ) 2 2 af ! j |0 ) (0 ,- 1 ) 2 2 ot 8 1 |0 ) (- 1 . 0 ) 2 2 5.3.1 Decomposition ofM according to duality It is useful to depict the Hilbert space H as in Figure 5.3. In the diagram, each line stands for a multiplet of states and each column contains iV-particle states. Note that the A st and B st operators of USp(4)s act within rows of the diagram, with the A st operators mapping iV-particle states to ( N + 2 )-particle states, and the B st operators acting in the opposite direction. The Cki operators act within columns of the diagram. Figure 5.3: Energy level diagram for H <2 2 >v x (0 0 )s ------ (21)vx(10)s __ ------ (2 0 >y x (1 1 ) 5 ------ (H)v x (2 0 ) 5 ------ (1 0 )v x (2 1 ) 5 ------ (0 0 )y x ( 2 2 )s ------ N = 0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 N = 8 The schematic diagram for El illustrates the group dualities present in the following Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2D) x USp(2w) D u a l i t y 72 network, with typical irrep labels placed next to the group to which they belong. 7r 0(16) x 0 (1 ) 7T U n { ! " } U(8 ) X U (l) {N} U n (5.29) M U(4) x U(2) u n (A) USp(4)y X USp(4)s (A*) Groups on the same line of (5.29) are dual in their actions on H. In the general situation, the network of groups that illustrates the duality relation ships, together with the form of the infinitesimal generators, is as follows: 0(4Qw) x 0 (1 ) U n a{paig U(2flo;) x U(l) Es Css U n (5.30) Er a k r a lr U ( 2 f l) X UM Cst u n Cki USp(2Q)y x U S p M s -Tst) B st, Cst The entire Hilbert space H carries a representation of 0(16). Since the infinitesimal generators of 0(16) that change the number of particles create or annihilate pairs of particles, H decomposes into two irreps of 0(16), one for which each basis state has an even number of particles and one for which each basis state has an odd number of particles. The two irreps are distinguished by the parity quantum number of 0(1). Thus, separating the diagram for H into even-numbered and odd-numbered columns illustrates the duality 0(16) x 0 (1 ). The infinitesimal generators of U( 8 ), of the form a\paiq — \5 ki&pq, do not change the number of particles, and so act within columns of the diagram. The states in a column labelled by N carry the {1^} irrep of U( 8 ), and also the {N} irrep of U(l) (the U(l) number operator ]T)S Css simply counts the number of particles, which is N for each state in the column labelled by N). Therefore, each column in the diagram for H displays U(8 ) x U(l) duality. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2f2) x USp(2w) D u a l i t y 73 The U(4) operators, of the form J2r [ al rair ~ \& k ij , and the U( 2 ) operators Cst also act within columns of the state diagram. The decomposition of each column of states into irreps of U(4) x U(2) is according to fermionic unitary-unitary duality. For example, the N = 2 column decomposes as U(8)jU(4)xU(2) {11} ^ {2} x {11}© {11} x {2 } (5.31) The uppermost line in the N = 2 column stands for 10 states, which together carry the {2} irrep of U(4). Each of the states carries the one-dimensional irrep {11} of U(2). The 15 states for the middle line of the N = 2 column and the 3 states for the lower line can be arranged in a rectangular 6 x 3 array. Each column of 6 states (consisting of 5 states from the middle line and 1 state from the lower line) carries the {11} irrep of U(4) and each row of 3 states carries the {2} irrep of U(2). Upon restriction to USp(4)y, each copy of the irrep {11} of U(4) decomposes into a 5-dimensional copy of the irrep (11) (carried by 5 states from the middle line) and a 1-dimensional copy of the irrep (0) (carried by 1 state from the lower line). Thus, the decomposition of the state diagram into columns displays the dualities 0(16) x 0(1), U(8 ) x U(l), and U(4) x U(2 ). Finally, the decomposition of the state diagram into rows displays the duality USp(4)y x USp(4)s. The USp(4)y infinitesimal generators A st act along the rows of the state diagram, mapping states in a column N to states in the same row of column N + 2, annihilating states in columns 7 and 8 . The B st operators act similarly, but to the left, annihilating states in columns 0 and 1. The Cst operators map states of a single line of the diagram to states of the same line. Each row of the state diagram carries an irrep of USp(4)y x USp(4)s, as indicated by the labels at the left of the rows. Consider the second row of the state diagram, labelled by (10)y x (2 1 ) 5 . The states in the N — 1 column of this row carry 2 copies of the (10) irrep of USp(4)y, as do the states in the N = 7 column, and the states of the N = 3 column carry 6 copies of the same irrep, as do the states of the N = 5 column. The collection of all of these states can be arranged into 4 rows, where each column carries a copy of the (10) irrep of USp(4)y, and each row carries a copy of the 16-dimensional (21) irrep of USp(4)s. Other rows of the state diagram decompose analogously. To label the states in H according to their spin and isospin, one would in general use the appropriate branching rules; i.e., the USp(4)y {, SU(2)y branching rule to determine the angular momenta, and the USp(4)s I SU(2 )s branching rule to determine the isospin values. For low-dimensional cases one can often infer the values by simply counting dimensions. The energy level diagram for H in Figure 5.4 includes the angular momenta and isospins. Note that the values are symmetric about the N = 4 column, which displays particle-hole symmetry. Also note that because the A st and B st operators create or annihilate pairs of particles coupled to zero angular momentum, the values of J are constant along each row of the diagram. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 5. USp(2f2) x USp(2w)D u a l i t y 74 Figure 5.4: Energy level diagram for El, showing spin and isospin content <22)y X <00)5 T = 0 J = 2,4 rp __ 1 rp _ 1_ x (10)5 (21) v 7 _ 1 5 7 J _ 1 5 7 ° 2 ’ 2 ’ 2 ^ — 2 ’ 2 5 2 (20)y X (11)5 T = 0 T = 1 T = 0 J = 1,3 J = 1,3 J = 1,3 T = 1 T — 12 t — 11 T — I (10) v X (21)5 _ 2 *___ 2_’ 2 ~ 2 ’ 2 2 j ~=Y J - 1 ^ = 2 ^ — I (00)y X (22>5 T = 0 T = 1 T = 0,2 T = 1 T = 0 J = 0 J — 0 J = 0 J = 0 J = 0 iv = 0 iV = 1 N = 2 N = 3 iV = 4 iV = 5 iV = 6 N = 7 N — 1 In more detail, eight basis states for the iV = 1 column can be arranged schematically as in the following array <^3 i |0) ®3l|0) 2 2 2 2 ai_l|0 ) 4 1 10) 5 5 5 5 (5.32) 4i_i|o) 4nlo) 2 2 2 2 «Ls_x |0 > o 4 i|0 ) 2 2 2 2 The Cki operators act within columns of the previous schematic diagram and the Ci_i operator acts to the right within rows. The states in each column span an J = | irrep of SU(2)^, and the states in each row span a T = | irrep of SU(2)s. Each of the lines in the Hilbert-space diagram can be depicted analagously. One notes from the Hilbert space diagram that quantum numbers for angular momen tum, isospin, and particle number N are insufficient to uniquely label all the basis states in EL The decomposition of El according to duality allows one to uniquely label states, since states w ith identical T, L, and N quantum numbers belong to different irreps of USp(4)y x USp(4)s. 5.3.2 A simple proof of the duality theorem Although a proof of the duality theorem is found in the original paper by Helmers [63], it is instructive to consider an intuitive proof for the low-dimensional example of this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2Q) x USp(2w) D u a l i t y 75 section. For instance, the following argument is intended to make plain Helmers’ rule for the Complementarity of Young diagrams occurring in the duality theorem. The goal of the argument is to show that common highest-weight states (i.e., states that have highest weight w ith respect to both USp(4)y and USp(4)s) must have a par ticular form. To this end, it is convenient to introduce the following type of schematic diagram to illustrate the V-weight and S-weight of a state, and place an “X ” in each cell corresponding to an occupied single-particle state. (5.33) Each cell in the diagram that contains an X means that the corresponding factor of od is present in the expression for a basis state. An unfilled cell means that the corresponding factor of a) is not present. In other words, cells containing Xs represent particles and unfilled cells represent holes. Since we are working w ith fermion operators, a cell can be filled with no more than one X. For example, the diagram 1 2 X 1 XX 2 3 1 1 3 2 2 2 2 corresponds to the state a 3 _ io ii« 1. 3 _i|0 ). 2 2 2 2 2 2 To obtain the S-weight and V-weight of a state, we refer to the expressions for the weight operators Css in equation (5.12) and Ckk in equation (5.17). To determine the first component of the S-weight of a state, subtract 2 from the number of Xs in the first row of the diagram. To determine the second component of the S-weight, subtract 2 from the number of Xs in the second row. To determine the first component of the V-weight, subtract the number of Xs in the first column from the number of Xs in the fourth column. To determine the second component of the V-weight, subtract the number of Xs in the second column from the number of Xs in the third column. For example, in diagram (5.34), the S-weight is (—1,0) and the V-weight is (0,1). Which states are common highest-weight states? The key point is that for a com mon highest-weight state, the entire right half of the diagram (i.e., the third and fourth columns in this case) must be filled with Xs. For if there is an unfilled cell in the third or fourth column, then one can add a pair of Xs, one in the unfilled cell in the third or fourth column, and another in a corresponding cell in the first or second column, so that the V-weight remains unchanged but the S-weight is increased. For example, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2D) x USp(2 following state 1 2 XX 1 2 X 3 1 1 3 2 2 2 2 has V-weight (2 , 1 ) and S-weight (0, —1). It cannot be a common highest weight state, because by adding a pair of particles coupled to zero angular momentum as follows 1 2 XXX 1 (5.36) 2 XX 3 1 1 3 2 2 2 2 we obtain a state with the same V-weight, but with a higher S-weight of (1,0). Thus, a necessary condition for a common highest-weight state is that the right half of the diagram be filled. Further consideration shows that if there are any filled cells in the left half of the diagram, then they must be pushed as far as possible upwards and to the right in order that the state be a common highest-weight state. For example, the following state, which has S-weight (1,1) and V-weight (1,1) 1 2 X XX 1 (5.37) 2 X XX 3 1 1 3 2 2 2 2 is not a common highest-weight state. The S-weight can be increased to (2,0) (keeping the V-weight the same) by moving the X in the lower left cell upwards: 1 2 XXXX 1 2 XX 3 1 1 3 2 2 2 2 Similarly, the V-weight of the state (5.37) can be increased to (2,0) (keeping the S-weight the same) by moving the X in the lower left cell of the original state to the right: 1 2 XXX 1 (5.39) 2 X XX 3 1 1 3 2 2 2 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2fl) x USp(2w) D u a l i t y 77 Pursuing the analogous reasoning in the general case results in the following pre scription for obtaining all common highest-weight states: A diagram corresponds to a common highest-weight state if and only if each of the following conditions is satisfied. 1. The right half of the diagram has all cells filled w ith Xs. 2. When the left half of the diagram is reflected about a vertical line, the occupied cells form a regular Young diagram. In general, the S-weight of a common highest-weight state is the partition corre sponding to the Young diagram described in condition 2 above. The components of the V-weight of a common highest-weight state are given by the number of unoccupied cells in the columns of the left half of the diagram. This completes a simple proof of Helmers’ theorem on the complementarity of the Young diagrams of paired irreps for the USp(2Q)y x USp(2w)s duality. Finally, to obtain an explicit expression for a common highest-weight state in general, one simply includes the appropriate factor of a t for each filled cell of the corresponding schematic diagram, as indicated above. To w it, given an irrep (/i) x (u) for the dual pair of groups USp(2fil) x USp(2w), draw the schematic diagram with w rows and 2CI columns. Fill each of the cells in the Cl columns in the right half of the diagram completely with Xs. In the left half of the diagram, the right-most uK remaining unfilled cells in the K -th row are then filled w ith Xs. The completed diagram can then be used to explicitly construct common highest-weight states. 5.4 Applications of USp(2fi) x USp(2u;)duality The work of Flowers [45] on the seniority classification of shell-model states in j j cou pling was extended by Helmers [63] to show the complementary relationship between seniority and quasispin quantum numbers. In Section 5.1.1 we discussed an example of this complementarity, the situation of nucleons of a single type in a single j-shell, where the relevant group duality is USp(2Q)y x USp(2)s, carried by the spin representation of 0(4f2), where Cl = j + | and ui = 2r + 1 = 1 . In Section 5.3 we discussed the example of neutrons and protons in a single jr-shell, with j = §, where the relevant group duality is USp(4)y x USp(4)s- Recall that USp(2)s is identical to the quasispin group SU( 2 )s. In the case of a single type of nucleon in a single j-shell, the quasispin and seniority quantum numbers are complementary, and the spectrum depends only on the seniority quantum number. In the more general situation, exemplified by the example of neutrons and protons in a single j-shell [62], the spectrum depends only on the quantum numbers of the generalized seniority group USp(2w)s', as shown in the diagram at the beginning of Section 5.3.1. This diagram also shows the value of duality in classifying states. Each of the lines in the diagram, which represents a multiplet of states, can be labelled uniquely Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 5. USp(2Q) x USp(2w)D u a l i t y 78 by quantum numbers of either of the two chains in the network of groups (5.29). A complete classification of each of the states w ithin such a m ultiplet can be achieved by labelling each state according to its angular momentum quantum numbers; for example, one can make use of the chain U(8 ) D U(4) D USp(4)y d SU(2) D SO(2) {1^} M (X) p J M where p distinguishes potential multiple occurrences of states of angular momentum J in a USp(4) multiplet. ■ Use of the chain (5.40) requires a determination of the USp(4) SO (3) branching rule. An effective algorithm, built on the Schuropera program, makes use of the following chain: USp(4) f U(4) | U(2) 4 SU(2) 4 SO(2) (A) {A/4} {3} % {A/4} = E„ <*<.{/■'} £ „«/. [{ Table 5.4: The branching rule USp(4) I SO(3) for some low-dimensional irreps Dimension USp(4) SO(3) 1 (0 ) J = 0 3 4 (1 0 ) J = 2 1 0 (2 0 ) J = 1,3 5 (1 1 ) J = 2 3 5 9 2 0 (30) J = 2’ 2’ 2 i 5 r 16 (2 1 ) J = 2’ 2’ 2 35 (40) J = 0 , 2 ,3,4, 6 35 (31) J = 1,2,3,4, 5 14 (2 2 ) J = 2,4 More complex pairing problems have been considered, where, for example, pairs of nucleons coupled to zero spatial angular momentum can have tota l spin angular momen tum S = 0 and total isospin T = 1, or S = 1 and T = 0. The group structure of such Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5. USp(2f2) x USp(2w) D u a l i t y 79 problems is reviewed in [155, §3.2] and further details are found in references therein, and in [150], [101], [42], and [112]. For further recent applications of USp-USp duality in the nuclear shell model, see [137, 124]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 S p ( n , R ) x O(N) duality Noncompact symplectic groups are fundamental in many areas of theoretical physics. The symplectic group Sp(n, R) is the most general set of linear transformations that preserve an anti-symmetric bilinear form, and such anti-symmetric forms occur often in physics .1 For example, the commutator is an anti-symmetric bilinear product, and the canonical commutation relations are at the foundations of quantum mechanics. Analagously, the Poisson bracket is an anti-symmetric bilinear product that lies at the foundations of classical mechanics. Thus, an important realization of Sp(n, R) is as the group of linear canonical transformations of a 2n-dimensional phase space; for classical models these transformations preserve the Poisson brackets, and for quantal systems they preserve the canonical commutation relations. Noncompact symplectic groups Sp(n, R) have important applications in descriptions of nuclear collective motion [126, 129]. The earliest algebraic model of nuclear rotations was the SU(3) model of E llio tt [39, 40]. This model and other im portant subsequent models of nuclear rotations and vibrations are submodels of the more general symplectic model of nuclear collective motion [132, 128], the basic observables of which span the Lie algebra sp(3, R). Many nuclear models are based on harmonic-oscillator potentials, since besides pro viding reasonable approximations to experimental data they frequently lead to significant simplifications in analytical solutions. The most common expression of the nuclear shell model is built on the harmonic oscillator. The single-particle states of the spherical harmonic oscillator carry the fundamental oscillator representation (comprising two ir reps) of the Sp(3, R) group, and so noncompact symplectic groups are of fundamental 1 We write Sp(n, R) to mean the noncompact symplectic group of rank n. This group is a subgroup of GL(2n, R) and has a natural realization in terms of 2n x 2 n matrices. In some of the literature this symplectic group is denoted Sp(2n, R). 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n,R) x 0(N) d u a l i t y 81 importance for nuclear theory. More generally, the Hilbert space spanned by the states of a single particle in an n-dimensional harmonic oscillator potential carries the fundamental oscillator represen tation 2 of the group Sp(n, R). Such representations are also known as Segal-Shale-Weil representations, Weil representations, harmonic representations, and metaplectic repre sentations. Oscillator representations of real symplectic groups are the arenas for the Sp(n, R) x 0 (N) duality discussed in this chapter, just as the fundamental spin repre sentations of the orthogonal groups are the arenas for the USp(2Q) x USp( 2w) duality discussed in Chapter 5. In symplectic models of nuclear collective motion, orthogonal groups O(N) act as symmetry groups for A/'-particle systems. That is, the H ilbert space H for the system decomposes into 0 (N ) irreps, where the states of each irrep are degenerate in energy. The Hilbert space H carries the fundamental oscillator representation of Sp(niV, R). The group Sp(n, R) acts as a dynamical group for the system, in the sense that states of H that span an irrep of Sp(n,R) are connected by operators that excite or de-excite collective motions. Because the groups Sp(n, R) and O (N) are dual in their actions on H, the states of H can be labelled by the quantum numbers of either group. Therefore, collective motions can be associated with particle-permutation symmetries, since the symmetric group Sat is a subgroup of O(N). This is important for the implementation of the symplectic model, and is discussed in Section 6.5.2. 6.1 Symplectic and orthogonal algebras Early models of nuclear collective models were phenomenological, and it was unknown for a long time how to interpret states of these models in microscopic (shell-model) terms. The symplectic model provides effective means for doing this, based on a strategy of formulating collective observables in algebraic terms [121, 122, 123]. Operators that are relevant for descriptions of nuclear rotations and quadrupole vibrations include the following ones, Qst,ij = XsiXfj (b-f) K st,ij — P siP tj (6.2) P st,ij = 2 (^siPtj Ptj%si) (6.3) technically, such an oscillator representation is an irrep of the covering group of Sp(n,R), which is known as a metaplectic group, symbolized by Mp(n,R). For the sake of simplicity, we have omitted technicalities about the metaplectic groups in this chapter; for more details, one can consult [64, 65] and [132]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 6. Sp(n,R) x O (N) d u a l i t y 82 which span an sp(niV,R) algebra. The indices s and t are particle labels, and range from 1 to N , where N is the number of nucleons in the system. The indices i and j label spatial directions and range from 1 to n, where n = 3 for the physical model. The Q operators are generalized quadrupole moments whose values describe configuration, the K operators are infinitesimal generators of momentum flows, and the P operators are infinitesimal generators of deformation and rotation of the system. By contracting on particle number, one obtains collective observables. The commutation relations of the Q, K, and P operators can be determined from the canonical commutation relations of the basic x and p operators, [xSi,ptj\= ih 5 st5ijI (6.4) To obtain suitable raising and lowering operators, one passes to the complex extension of the sp(nN, R) algebra in the usual way, by defining boson raising and lowering operators = t/W (*- “ ■ Ki = \IW{x-i + m,p-') (6-5) where M and u> are parameters associated with the simple harmonic oscillator Hamilto nian operator for the system H° = 2if 5 3 ^ + \ Muj2 (6-6) si si The commutation relations satisfied by the boson operators, [bsu b]^} = 5sA :jl (6.7) follow from the canonical commutation relations (6.4). In terms of the boson operators, a set of operators that spans the above-defined sp(n7V, R) algebra is = blfilj (6.8) Pst,ij ~ bSibtj (6.9) Cst,ij = 2 (blibtj + btjb\^j = b\jbtj + -Sst8ijl (6.10) The A operators create two quanta of energy and the B operators annihilate two quanta of energy. The noncompact nature of sp(nN, R) is reflected in the fact that an indefinite number of A operators can be applied to a state, so that the energy can be raised indefinitely. The C operators preserve the number of quanta, and they span a U(niV) subalgebra of sp (nJV, R). Before discussing duality relations, the following sections provide a brief summary of symplectic and orthogonal algebras and their irreps. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n, R) x 0(N) d u a l i t y 83 6.1.1 sp(n,R) algebras In this section we discuss the irreps of symplectic algebras. We in itia lly consider irreps for a single particle (N = 1), so that the operators (6.8-6.10) that span sp(n, R) reduce to An = b\b\ (6 .11) Bij - bibj (6.12) C,j = \ + w ! ) = b'h + I f t j l (6.13) where i and j are in the range 1 ,2,..., n. Using the canonical commutation relations for boson operators (6.7), one can derive the following sp(n, R) commutation relations [Ajj,A/;/] — 0 (6.14) [B ij,B ki] = 0 (6.15) [Cjj, A/./] = 8jkA.ii + 5jiAik (6.16) \Cij, B ki\ — SikBji 8n Bjk (6.17) [C^, Cki\ — bjkCu — 8uCkj (6.18) [Bij,Aki] = 8jkCn + SjiCki + 8ikCij + SuCkj (6.19) The Cij operators conserve the number of harmonic oscillator quanta and span the subalgebra u(n) C sp(n, R). The n weight operators for the sp(n, R) algebra are the Cu operators. For i < j , the Cij are raising operators, and for i > j the Cij are lowering operators. The A jj are all raising operators, and the Bij are all lowering operators. The unitary irreps of Sp(n, R) are infinite-dimensional. O f these, the physically most relevant irreps are those that occur in the space of some harmonic oscillator. Each such irrep has a unique lowest-weight state |cr) that is annihilated by all of the lowering operators. A lowest-weight state is uniquely labelled by the weight a = (cq, <72, . . . , cr„), where Cu\a) = and so the irreps can be uniquely labelled by the weights of their lowest-weight states. The single-particle states of an n-dimensional harmonic oscillator span the fundamen tal oscillator representation of Sp(n, R). Beginning with the vacuum state and acting on it repeatedly with all combinations of raising operators, one obtains an infinite number of states, all of which have an even number of quanta. These states span the (|(0 )) irrep 3 of Sp(n, R). Repeating this process w ith the one-quanta states &||0), where i = 1,2,..., n, one obtains an infinite number of states, all of which have an odd number of quanta. 3By convention, the labels for real symplectic groups are enclosed by angle brackets. The notation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n,E) x 0 ( N ) d u a l i t y 84 These states span an irrep 4 of Sp(n,M) that is symbolized by (|( 1)). The collection of all of these states, both even and odd, forms a basis for the oscillator representation of Sp(n, E), which is symbolized ( 5 (0 )) © (|(1)). Multi-particle states of an n-dimensional harmonic oscillator also belong to irreps of Sp(n, E). Specifically, if-particle irreps can be obtained by decomposing if-fold tensor powers of the fundamental oscillator representation. Such irreps all have lowest weights of the form (y + oq, y + a 2, ..., y + on), which is written as (y (<7)). The Hilbert space HI that carries the oscillator representation of Sp(ra, E) can be decomposed into a direct sum of subspaces, H — E ” =o , where each subspace carries the irrep {N} of U(n). Physically, one notes that the lowest-energy state in the subspace of H that carries the (|(0)) irrep of Sp(n,E) (that is, the vacuum state) has energy \ftw. The energy of the states in the lowest-energy multiplet for (|(1)) is fftw, and in general for (y (a )) the lowest energy is ( if/2 + \o\)Tujj, where a- is a partition of \a\. 6.1.2 o(N) algebras Since orthogonal transformations commute with simple harmonic oscillator Hamiltonian operators, orthogonal groups act as symmetry groups for systems described by simple harmonic oscillators. The Lie algebra 0 (N) is spanned by the operators L st = ^ (xspt - xtps) (6 .21) where s and t are in the range 1 ,2 ,..., iV. The Lst operators have commutation relations \L$ti L^i] = i {StkLgi 6 tiLsk ^sk^ti T SgiLtk) (6.22) To obtain raising, lowering, and weight operators, one passes to the complex extension by defining o (N) operators in terms of N boson creation and annihilation operators [131]: Lst = - i (b\bt - b\bs) (6.23) Now suppose that N = 2M is even. By analogy with the construction of raising and lowering operators for so(3), one can define the following operators [131] in terms of those of equation (6.23): L p + , q ± = i (L 2 p ,2 q~ 1Lzp-lfiq) ± ( L 2p t2 q1 - — iL 2p-l,2q-l) (6.24) L p - , q ± = i { L 2 p ,2q + il/ 2p -l, 2g) i (L 2pt2q- i + iL2p_l, 2q -l) (6.25) 4Technically, this is a projective representation of Sp(n, K) but a true irrep of the covering group Mp(n, R), in the same way that SO(3) has spinor irreps that are true irreps of SU(2). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 6. Sp(n, R) x 0(N) d u a l i t y 85 where p and q range over 1,2,..., M. Then one can identify raising, lowering, and weight operators through the definitions 1 1 1 1 •Apq = — Lp+ The Apq are raising operators and the Bpq are lowering operators. Among the CPq are raising operators (for p < q), lowering operators (for p > q), and weight operators (for p = q). I f N is odd, one must include the additional operators Lp+,n = i (L2P,n — i i 2p-i,Ar) (6.27) Lp~,n — i (L2P,n + i-^p-i.iv) (6.28) Then Ap = Lp+tN are raising operators and Bp = LP-tN are lowering operators. Irreps of O(N) are labelled by their highest weights, just as for other compact Lie groups. Since 0 (N) has rank M , where either N = 2M or N = 2M + 1, the highest weights of 0 (N) irreps are of the form (cu, cr2, • ■ •, 02 > • • • > &M > 0- 6.1.3 Symplectic and orthogonal subalgebras of sp(niV, R) From the sp(niV,R.) operators (6.8-6.10), one can form operators that span an sp(«, ‘ subalgebra of sp(niV, R) by contracting on the particle-number indices: N Aij — bsibsj (6.29) s—1 N B%j — ^ ^ bsibsj (6.30) £-1 N j y C i j = b \jb sj + — (6.31) Taken by themselves, the operators in (6.31) span a u(n) subalgebra of sp(n, R). Beginning again with the sp (niV, R) operators, one can form operators that span an o(N) subalgebra of sp(niV, R) by contracting on spatial indices: n Lst = - iY ,{ blib» - btib” ) (6-32) «=1 One obtains raising, lowering, and weight operators as in the previous section. One can show directly that the sp(n, R) operators as constructed in equations (6.29- 6.31) commute with the o(N) operators of equation (6.32). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 6. Sp(n, R ) x 0(N) d u a l i t y 86 6.2 The Sp(n, R) x O(N) duality theorem Consider the H ilbert space of N particles in an n-dimensional harmonic oscillator potential. This space carries the fundamental oscillator representation of Sp(niV,K). The actions of Sp(n, R) and 0 (N) commute on WN, and thus form a direct product of groups. Furthermore, decomposes as a “m ultiplicity-free” direct sum of irreps of Sp(n, R) x O (N); that is, each irrep of Sp(n, R) x O(N) that appears in the direct sum appears only once. Finally, the irreps of Sp(n, R) and O (N) that appear in the decomposition are in one-to-one correspondence, so that Sp(n, R) and O (N) form a dual pair of groups in their actions on HA'. The explicit correspondence between the symplectic and orthogonal irreps in this symplectic-orthogonal duality theorem can be expressed as 5 [80] (6.33) where the sum extends over partitions A for which Ai + A2 < N and Ai < n (6.34) The symplectic-orthogonal duality theorem (6.33) was first stated in this form in [74]. The constraints (6.34) were proved in [141] to include each unitary irrep of Sp(n, R) that has a lowest weight. Among the duality relations discussed in this thesis, symplectic-orthogonal duality is the only one that includes infinite-dimensional irreps of noncompact groups. This duality relationship was first noted by Lohe and Hurst [100] and Moshinsky and Quesne [109]. An early proof of the Sp(n, R) x O(N) duality relation can be found in Chacon [22]. Later Howe generalized these and other works and developed the concept of dual reductive pairs of subgroups of noncompact symplectic groups; see [64] and references therein. 6.3 Example: Sp(l,R) x 0(3) duality on the space of the Sp(3,R) oscillator representation A physical interpretation of this example is that of 3 particles in a one-dimensional harmonic oscillator potential. As such, the generators of Sp(3, R) are only required to 5Unitaxy irreps of orthogonal groups are labelled by their highest weights. It is conventional to enclose such labels by square brackets for orthogonal groups. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(ra,R) x 0 (N) d u a l i t y 87 have one index: -n-st — uswtbibJ (6.35) Bst = bsbt (6.36) Cst — b\bt + -5 stl (6.37) where s and t are in the range 1,2,3. Contracting on the particle index results in the S p(l,R ) generators: A ^ b\b\ (raising operator) (6.38) S B bsbs (lowering operator) (6.39) 5 2 C + xl (weight operator) (6.40) 5 The 0(3) generators are of the form L st = - i (b\bt ~ bibs') (6.41) The weight operator is L0 = T12, and the raising and lowering operators are L± = 1/23 i iT^3i. A straightforward calculation shows that each of the Sp(l,R) generators commutes with each of the 0(3) generators. To construct irreps of Sp(l,R) x 0(3) it is helpful to define the following operators: b{ = b \± i b\ b± — h =F i &2 (6.42) whence [&+,&+ = b -,b l = 21 and 6-, 6+1 = [6+,6L (6.43) The highest-weight state for an 0(3) irrep labelled by [/] is ( * t ) ‘ |0> (6.44) The remainder of the states that span the irrep are obtained by acting repeatedly on the highest-weight state with the 0(3) lowering operator, L_. Since the Sp(l,R) weight operator C commutes w ith L_, all of the states in an 0(3) irrep have the same S p(l,R ) weight. The 0(3) highest-weight states (6.44) are also Sp(l,R) lowest-weight states, and so we call them common extremal states. One can see that the states (6.44) are Sp(l, R) lowest-weight states because B (b+ ) 1 |0) = 0 (6.45) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 6. Sp(n,R) x 0(N) d u a l i t y 88 which follows from the commutation relations B , b \ ] = 2 Y +b_ and 6 ^ 1 = 0 (6.46) The Sp(l,R) weights of the states spanning the [/] irrep of 0(3) follow from the commutation relation [c, = i ( * t ) ' (8.47) which can be derived from [C, 6 +] = b\. Thus, each basis state in the [/] irrep of 0(3) has S p(l,R ) weight (I). One can therefore construct an Sp(l,R) irrep labelled by (§(/)) by beginning with one of the states belonging to the [I] irrep of 0(3) and applying the A operator repeatedly. This shows the 1-1 correspondence between Sp(l,M) and 0(3) irreps. To complete a proof of the duality theorem in this simple case, one must show that every state in the oscillator irrep of Sp(3, R) belongs to one of the Sp(l,M) x 0(3) irreps. That is, one would like to show that every such state can be constructed by using the operators A, L_, and 6 + acting repeatedly on the vacuum state. Since 63 = —|[L _ , b\], X~bl_ = [L-,b\], f>i = |(&+ 4- &L), and = |(fr+ ~ &-)> if follows that the Sp(3,M) raising operators A^ can be constructed from A, L -, and b+, and the proof follows. The entire Hilbert space of the oscillator representation of Sp(3, K) can be represented in a schematic diagram that illustrates the Sp(l,R) x 0(3) irreps as follows: Figure 6.1: Hilbert space that carries the oscillator representation of Sp(3,M) N = 5 ______ N = A ______ N = 3 ______ N = 2 ______ N = 1 ______ N = 0 ______ 1 = 0 ■' 1 = 1 1 = 2 1 = 3 1 = 4 1 = 5 Each line in the diagram stands for a multiplet of 21 + 1 states, and the states extend upwards indefinitely. There are a number of group dualities at play here. The states in each column transform among themselves with respect to both Sp(l,K) and 0(3) transformations; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n,R) x 0(N) d u a l i t y 89 both symmetries are labelled by the quantum number I. The states in each row transform among themselves with respect to both U (l) and U(3) transformations; both symmetries are labelled by the quantum number N. Finally, all of the states in all of the odd- numbered columns are odd-parity (w = —1) states, and all of the states in all of the even-numbered columns are even-parity (n = +1) states. States w ith the same parity transform among themselves with respect to both Sp(3,R) and 0(1) transformations. The following network of groups summarizes the duality relationships for the spherical harmonic oscillator. 7r Sp(3, R) x 0 (1) 7r U n N U(3) x U(l) N U n I 0(3) x S p(l,R ) I As w ith all group dualities, the basis states of each subspace in a dual-group decom position can be placed in a rectangular array that clearly illustrates the duality. For example, the I = 1 column of the spherical harmonic oscillator diagram, where each line represents 3 basis states (21 + 1 = 3), can be drawn w ith all basis states explicitly shown: N = 7 ______ N = 5 ______ N = 3 ______ N = 1 ______ 1 = 1 1 = 1 1 = 1 m = — 1 m = 0 m = 1 Once again we see the rectangular array of states characteristic of group duality, where rows span isomorphic copies of the I = 1 irrep of SO(3), and columns span isomorphic copies of the I = 1 irrep of Sp(l, R). The general symplectic-orthogonal duality is represented by the following network of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 6. Sp(rc,R) x 0(A) duality 90 groups. 7T Sp(nA, R) X 0(1) 7r U n {k } U (nA ) X U (l) {k } U n {A} U(JV) X U (n) {A} u n M 0 (N ) X Sp(n, R) (<*> A physical interpretation of this duality involves A particles in an n-dimensional har monic oscillator potential. A schematic diagram for the states of the Hilbert space (i.e., that carries the oscillator representation) for this case resembles the one for the 3-dimensional harmonic oscillator; the difference is that each line in the diagram for the n-dimensional case typically represents many more states than for the 3-dimensional case. The entire H ilbert space decomposes into two irreps of Sp(nA, R) x 0(1), labelled by parity. As before, states in the odd-numbered columns belong to one of the irreps, and states in the even-numbered columns belong to the other. Each row of the diagram represents a kfru energy level. The states of the k-th row carry the irrep {&} x {k } of U (n A ) x U ( l) , which decomposes into irreps of U (A ) xU (n ) according to bosonic unitary- unitary duality. Finally, each column carries an irrep (a) x [a] of Sp(n, R) x 0(A ), and so the decomposition of the diagram according to columns illustrates symplectic-orthogonal duality. 6.4 Common extremal states for Sp(n, R) x O(N) du ality To perform concrete calculations, one often needs to explicitly construct states. For the general symplectic-orthogonal duality Sp(n, R) x 0(A ), we do this by constructing common extremal states; that is, states that have highest weight w ith respect to 0 (A ) and lowest weight with respect to Sp(n, R). In general, the operators ( and are defined as ^ (6.50) ®fe±,z = hk-i,i T ihk,i (6.51) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n, R) x O(N) d u a l i t y 91 and satisfy the commutation relations = an<^ = = 25pk5ql (6.52) Then define the symbol \K\ as a K x K determinant with ij-element equal to B|+ ^N_j+1y For example, 111 sl + , i V (6.53) ® 1 + , IV |2| = ® l + , ( j V - l ) (6.54) ® 2 + , jV ® 2+ ,(JV—1) ®1+,JV ®l+,(iV-l) B l + ,(JV—2) (6.55) |3| = ®2+,jV ®2+,(A-l) ® 2 + ,(N-2) ® 3 + ,J V ®3+,(jV-l) ®3+,(JV-2) and so on. Then the following state is a common extremal state for both Sp(n, R) and O (N), with common weight (pi, a2, ■.., crK): where K satisfies the constraints of the duality theorem given in equation (6.34). By construction, C\ > a 2 > ■ ■ ■ > c?k - First let us show that the state in equation (6.56) is a weight state for both Sp(n, R) and 0(N ). Using the expressions in equations (6.31) and (6.50), straightforward calcu lations show that Cij,M\+l (6.57) and it follows that Ca,: k+ ,1 = 6i, (6.58) and so [Cih \K\R] = R\K\r ( iii > N - K + l) (6.59) [Cu,\K\r ] = 0 (if i < N — K + 1) (6.60) Therefore, C um = a im (6.61) and so the state |$) is a weight state for Sp(n, M) with weight (a). To establish that |<3>) is a weight state for 0 (N ), we begin with the commutation relation (6.62) Cpq,'PQ^k+ b! ,1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n,R) x 0(N) d u a l i t y 92 which is derived using the expression for Cn obtained from equations (6.26), (6.24), and (6.32). The action of the weight operators is then -TP> mk+,l 'pk^k+,1 (6.63) from which it follows that [ C ^ l K f ] = R\K\r (if p and thus Cpp\$) = crP|$) (6-66) The state |$) is therefore a weight state for O (N) w ith weight (cr). To show that |$) is a lowest-weight state for Sp(n, R), we must show that all of the latter’s lowering operators annihilate the state. A straightforward calculation using equation (6.30) shows that -By, — SjiBk- ,j + j (6.67) where the operators have been defined in equation (6.51). From this it follows that the action of By on \K\ produces a linear combination of K x K determinants, in each of which an entire column of B* elements has been transformed into B elements. One can conclude from this (making use of equation (6.52)) that By|$) = 0 (6 .68 ) For operators of the form Cy, using equation (6.57) one can show that, for instance, ®1+,IV JD>l+ ,(iV -l) ®l+,i 1+ ,(A—1) S1+,A 1 +,« Cij, 64'jN (6.69) i t ^2 +,N ® 2 + , ( A - l ) ®2+,j ®2+,(A-l) ®2 +,N In general, the action of Cy on the determinant \K\ is to replace column subscripts j by i, provided that j > N — K + 1. This results in a determinant equal to zero, if i > j, since two columns are identical. If j < N — K + 1 then Cy commutes with \K\. In either case, one can deduce that for lowering operators Cy C y|$) = 0 f o r i > j (6.70) and therefore |$) is a lowest-weight state for Sp(n, R). Now to show that | = 0 (6.71) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n,R) x 0(A ) d u a l i t y 93 and ■'pgiMJ,k+ ,i (6.72) If N is odd, then one uses equation (6.27) to show that additionally = 0 (6.73) The commutation relations of the previous paragraph make clear that the A operators annihilate |$). The action of a Cpq raising operator (for which p < q) on the determinant |if | is to decrease the first subscripts in an entire row from q to p (provided that q < if ) , which results in the determinant vanishing, since two rows would then be identical. If q > i f , then Cpq commutes with |if |. In summary, the Cpq raising operators also annihilate |$), and this shows that| o\ The first constraint is that the Young diagram has n rows or fewer, which amounts to K < n. This is clear from the structure of the determinants that form | 6.5 Applications of Sp(n,R) x 0(A) duality Symplectic-orthogonal duality has applications in a number of areas of physics, a dis cussion of some of which is found in [138, §VII]. For instance, in beam optics the action of an active optical element on the spatial state of a beam is a symplectic transforma tion, and the action on the polarization state of a beam is an orthogonal transformation. The presence of symplectic-orthogonal duality in this case could facilitate the analysis of multi-mode interferometry. Another application of symplectic-orthogonal duality is in the resolution of certain state-labelling problems. In applications of group representation theory to nuclear physics, a useful way to label basis states of a group representation is to determine a chain of subgroups of the group and to identify the irrep of each subgroup that the basis state belongs to. For unitary groups, there is a chain of subgroups for which this procedure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n,R) x 0(N) d u a l i t y 94 provides sufficient labels to uniquely specify each basis state. However, the chain of subgroups used, U(n) D U(n - 1) D U(n - 2) D • • O U (l) (6.75) is typically not physically relevant, and so the use of such a basis entails the calculation of change-of-basis transformations to physically relevant bases. It is more natural to use a subgroup chain that is physically relevant, so that the quantum numbers labelling basis states in a basis adapted to the chain have physical meaning. In many such situations, the number of labels generated in this way is insufficient to uniquely specify each state. This is known generically as a state-labelling problem [130]. A resolution of the state-labelling problem for the important case U(n) D 0 (n) by Le Blanc and Rowe [89, 90, 91], and also by Deenen and Quesne [34, 116, 117, 118, 119], makes essential use of symplectic- orthogonal duality. The remaining sections of this chapter discuss applications of symplectic-orthogonal duality to the separation of nuclear motions into collective and intrinsic components, and to the corresponding classification of states. 6.5.1 Collective and intrinsic motions in nuclear collective mod els Applications of symplectic-orthogonal duality in nuclear collective motion are reviewed in [128] and [132]. Beginning in the 1960s, an approach to nuclear collective motion based on orthogonal groups was pursued by researchers in the former Soviet Union [132, §8.3], based on a recognition that 0 (A ) is a symmetry group for a nucleus w ith A nucleons. Specifically, the relevant chain of orthogonal groups is 0 (3 A ) D 0(3(A - 1)) D SO(3) x 0 (A - 1) (6.76) In this approach, collective model spaces are subspaces Hem of translationally-invariant shell model states in which every state shares a common 0 (A — 1) wave function. The orthogonal groups provide good quantum numbers for the labelling of states, and also act as symmetry groups since their operations commute with the relevant Hamiltonian. However, there are two difficulties w ith this approach. First, there is no way using only these orthogonal groups to distinguish states with the same quantum numbers belonging to different subspaces that carry equivalent 0 (A — 1) irreps. Second, only the states of the subspaces Hen that have the correct permutation symmetry are contained in the nuclear Hilbert space. That is, the nuclear Hilbert space is not invariant with respect to all 0(A — 1) transformations. It was later observed that the collective model subspaces are carrier spaces of Sp(3, R) irreps, and that Sp(3, R) acts as a dynamical group that is dual to the orthogonal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 6. Sp(n,R) x O (N) d u a l i t y 95 symmetry group. Since the carrier spaces of Sp(3, R) irreps lie completely within the nuclear H ilbert space, the Sp(3, R) operators can be used to diagonalize collective model Hamiltonians. This m ajor advance is embodied in the symplectic model of Rosensteel and Rowe [?, 122, 123]. But the following problem then arose: How does one combine spatial wave functions with spin/isospin wave functions to obtain antisymmetric total wave functions? This could be achieved if one could identify the permutation symmetries associated w ith particular symplectic irreps. Symplectic-orthogonal duality allowed Rowe, Rosensteel, and others to use both sym plectic and orthogonal groups to advantage in describing nuclear collective motion. One can identify the relevant many-particle subspaces of the nuclear H ilbert space as Sp(3, R) irreps, and then one can identify their associated permutation symmetry by making use of the duality Sp(3A, R) D Sp(3, R) x 0(A) and the branching 0(A) I Sa - This is discussed in detail in the following section. As an additional benefit, the use of symplectic-orthogonal duality allows one to natu rally separate nuclear motions into collective and intrinsic components. Consider a single particle in a three-dimensional harmonic oscillator potential. The relevant dual chains of groups are given in (6.48). Wave functions that form a basis for the Hilbert space in this situation can be written in product form, as tpnim(r, 9 ,4>) = R ni{r)Yim(9,4>) (6.77) For a particular value of I, the radial wave functions Rni(r) transform among themselves with respect to Sp(l,R) transformations. Also, for a particular value of I, the spherical harmonics Y im (9, 4>) transform among themselves with respect to SO(3) transformations. This one-to-one correspondence between the radial and angular symmetries, labelled by the common quantum number /, indicates symplectic-orthogonal duality between the groups Sp(l,R) and S0(3). Remarkably, the same type of duality that holds for this separation of a single-particle wave function into radial and orbital factors also holds in the multi-particle case. S0(3) acts as a symmetry group for the system, since all of the group’s transfor mations commute with the system’s Hamiltonian operator. The energy of the system depends only on n, and so the dynamics of the system is described by S p (l,R ), which is thereby considered to be a dynamical group for the system. A physical consequence of symplectic-orthogonal duality is that the configurational symmetry and the dynamics of this system are in one-to-one correspondence. Another way to say this is that the radial and orbital motions of the system are interlaced, so that once one is specified the other is immediately known. Remarkably, the same sort of interlacing also exists in the analogous multi-particle situation. Consider N nucleons in a 3-dimensional harmonic oscillator potential. The relevant symplectic-orthogonal duality, Sp(3, R) x 0(A ), with the groups acting on the space of the oscillator representation of Sp(3A, R), can be interpreted in terms of col lective and intrinsic motions in the nucleus. Specifically, an irrep of Sp(3, R) is carried Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 6. Sp(n, M) x 0(N) d u a l i t y 96 by a space of collective states of the nucleus, whereas spaces that carry irreps of the corresponding dual group 0 (N ) describe intrinsic motions. It is interesting to contrast the role of 0 (N ) in multi-particle symplectic-orthogonal duality with the role of SO(3) in the single-particle case. In the latter case, SO(3) symmetry relates to the geometry of the particle’s three-dimensional configuration space. In the multi-particle case, 0 (N ) acts on an abstract “particle-index space,” and the orthogonal symmetry in this case is a generalization of particle-permutation symmetry. The same mathematical structure applies to two distinct physical situations. 6.5.2 Classification of states in the symplectic shell model In the symplectic shell model [128, 132], one uses a basis for an T-particle nucleus that reduces the chain S p(3^,E ) x U(4) D Sp(3,R) x 0(,4) x U(4) (6.78) The reduction continues as in the following equations: Sp(3,M) D U(3) D SU(3) D SO(3) D SO(2) (6.79) O(yl) D SA (6.80) U(4 ) D SU(4) D SU(2)s x SU(2)t D U ( l) s x U(1)T (6.81) The irreps of the group SO(3) are labelled by angular momentum, 0(^4) and are involved with the permutation symmetry of the A nucleons, and U(4) is W igner’s su- perm ultiplet group that is used to describe the spin and isospin of individual nucleons. Ultimately, a wave function that describes the entire system must be antisymmetric w ith respect to particle permutation symmetry, so the irrep labels of individual groups must be compatible w ith this constraint. Classification of basis states based on W igner’s su- perm ultiplet group are discussed earlier in this thesis, and so here we focus on how one classifies the Sp(3,R) irreps that occur in the space of the nuclear shell model for a par ticular nucleus. That is, in the branching of the fundamental oscillatorrepresentation of Sp(3^4,R) to its subgroup Sp(3, R) x 0(^4), Sp(37l,R) | S p (3 ,R )x O (^ t) (6.82) ( \ m + < ](i» + <6-83) CT one would like to know the particle-permutation symmetries associated w ith the Sp(3, R) irreps, which amounts to the further branching Sp(3, R) x 0(^4) 4. S p (3 ,R )x S ^ (6.84) X ^ 4- ^ ^ au{]-A(a)) x {v } (6.85) <7,V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 6. Sp(n,R) x 0 (N) d u a l i t y 97 This problem has been solved by Carvalho [15] and further developments have been made by Wybourne [162]. It is equivalent, by duality, to determining the plethysms of the yl-th power of the fundamental oscillator representation of Sp(3, R): (<|(0)) + <|(l»y =£xs(4(°)>+ < !(!)> ) © M (6.86) where Xe is the dimension of the corresponding irrep. For a particular irrep {^A{a)) of Sp(3, R), one can identify the associated particle per mutation symmetries {i/}, and the multiplicities caLn simply by making use of the known 0(^4) I Sa branching rule [162], Conversely, one needs to determine which Sp(3, R) irreps are associated w ith a particular permutation symmetry. This is again achieved by making use of the 0(^4) 4- S ,4 branching rule. One simply determines which 0(^4) irreps branch to the specified irrep, and by duality this determines the corresponding Sp(3, R) irrep. Since there are an infinite number of such irreps, one must be satisfied by specifying a cut-off energy level. For the case of four particles, the relevant portion of the 0(4) 4- S 4 branching rule table (for a cut-off at the Ahu energy level) is given in Table 6.1 (adapted from page 4392 of [162]): Table 6.1: The 0 (4 ) 4- S4 branching rule for some low-dimensional irreps 0(4) s 4 [0] {4 } [1] {4 } ® {31} [2] {4} ©2(31} ©{22} [11] {31} ©{211} [3] 2{4} © 3{31} © {22} © {211} [21] 2{31} © 2{22} © 2{211} [4] 3{4} © 4{31} © 2{22} © 2{211} [31] {4} © 4{31} © 2{22} © 4{211} © { l4} [22] {31} © 2{22} © {211} Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 6. Sp(n,M) x 0(N) d u a l i t y 98 Using Table 6.1, one deduces that (up to a cut-off at the 4 ftuj energy level) ((i(0)) © <|(D>) ©{4} = <|(0)> ® © {3 1 } = <|4(1)) ® 2 ©4(^(4)) © 4 < |(0 ))® < |(D > ) © {2 2 } = <|4(2)> ® <|4(3)> ® 2<|4(21)> ® 2<|4(4)> © 2<|4{31)> ©2(b(22)) (6.89) <|(0)) ® <|(1)>) © {211} = (14(11)) ®(i4(3)>® 2(14(21)) ®2 ©(14(22)) (6.90) ((l(0))® Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 Summary This thesis presents a physical perspective on group duality and discusses its physical ap plications, in particular those most relevant in nuclear physics. By presenting the theory of group duality and deriving the duality relationships in the language of many-particle quantum mechanics, we hope to make the abstract mathematics more accessible to physi cists. Conversely, we hope the insights gained from the concrete physical applications will provide a valuable new perspective to mathematicians. In this way further dialogue between physicists and mathematicians may be stimulated. The most familiar duality in the physics community is Schur-Weyl duality. One of the goals of the thesis is to bring to the attention of the physics community that group duality is more general than Schur-Weyl duality. Furthermore, although certain aspects of Schur-Weyl duality are “well-known,” other aspects have been “forgotten,” and it is the intention of the thesis to clarify the role of Schur-Weyl duality in symmetry analysis and its relationship to the other dualities. There are many important physical applications of group duality. The general theme of such applications is to translate a difficult problem involving a group or group chain into a simpler problem involving a dual group or dual chain, solve the problem, and then translate the solution back to the original group or group chain. Some such applications are: • decomposition of tensor products (Clebsch-Gordan coefficients) • determination of branching rules • construction of physically acceptable product wave functions • determination of plethysms • deriving the characters of one group from the known characters of its dual group 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 7. S u m m a r y 100 • deriving tensor product relations, branching rules, plethysms, etc. for a group or a group chain by duality from the analogous relations for a dual group or dual chain • labelling basis states using group chains based on dual sym metry/dynamical groups • diagonalizing Hamiltonian operators and determining matrix elements of relevant operators using dual symmetry/dynamical groups The group dualities discussed in this thesis can be classified into two types, those that apply to fermions (such as USp-USp duality) and those that apply to bosons (such as Sp-0 duality). There are two varieties of unitary-unitary duality, one of which applies to fermions and the other to bosons. Furthermore, a H ilbert space that carries Sp-0 duality can be decomposed into a direct sum of subspaces, each of which carries bosonic unitary-unitary duality. Similarly, a Hilbert space that carries USp-USp duality can be decomposed into a direct sum of subspaces, each of which carries fermionic unitary- unitary duality. Schur-Weyl duality applies to any type of particle, and underlies the other dualities in the sense that one can use Schur-Weyl duality to prove each version of unitary-unitary duality. Schur-Weyl duality deals with the physical situation of combining many particles into one 7V-particle system, where each particle is described by the fundamental irrep of U(n). The Schur-Weyl duality theorem states that the iV-particle H ilbert space decomposes into subspaces, each of which carries a single irrep of U(n) x SN. The irreps of U(n) and Sjv that appear are in one-to-one correspondence, and so each can be identified by a particle-permutation symmetry, specified by an irrep of the symmetric group Sat. If each of the individual particles is described by an irrep of U(n) that is not the fundamental irrep, then Schur-Weyl duality can be applied to a larger space so that one can still decompose the iV-particle H ilbert space into subspaces called plethysms, which again are characterized uniquely by particle-permutation symmetry. A new method for calculating plethysms has been explained in Chapter 4. Some of the most important computations of group representation theory applied to physics, such as the decomposition of tensor product representations and the determina tion of branching rules, are based on Schur-Weyl duality. The two unitary-unitary dualities are relevant for the problem of classifying product wave functions for a particle, where each factor wave function describes a different at tribute of the particle. For instance, a total wave function for a particular particle may be formed from the product of a spin wave function and an isospin wave function. The symmetry properties of the total wave function depend on the symmetry properties of the factor wave functions, and unitary-unitary duality specifies the precise relationship. O f the two other dualities that have been discussed, USp-USp duality is relevant for fermionic systems, and has applications in descriptions of nuclei involving pairing interac tions. Sp-0 duality is relevant for bosonic systems, and has applications in descriptions of nuclear collective motions. For each of these types of duality, the identification of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 7. Su m m a r y 101 symmetry groups and dynamical groups in the dual chains of groups is interesting. In the fermion case, the dual group chain is (from equation (5.30) of Chapter 5) 0(4 flw ) X 0 (1) U n akpalq U(2Qw) X U (l) S s C„ U n (7.1) Ylr akralr U (2fl) X U(w) Cst U n Cki U Sp(2fl)y X USp(2w)s ■A-sti Bst 'St For a Hamiltonian that includes pairing interactions, the groups in the chain U(20w) D U(2Q) D USp(2fl)y preserve the number of particles and act as symmetry groups, whereas the groups in the chain U (l) C U(w) C USp(2w),s act as dynamical groups. In the boson case, the dual group chain is (from equation (6.49) of Chapter 6 ) 7T Sp (niV,M) X 0 ( 1) 7T U n {k } U(niV) X U (l) {*} U n (7.2) {A} U (N) X U(n) {A} U n M O(N) X Sp(n, R) (ff) For a collective motion Hamiltonian, the groups in the chain U(niV) d U (N) D 0 (N) preserve the number of particles and act as symmetry groups, whereas the groups in the chain U (l) C U(n) C Sp(n,R) act as dynamical groups. 7.1 A brief guide to literature on duality In this section we present a brief historical overview of research on group duality. The comments in this section are intended to orient interested researchers to the literature, and do not constitute a comprehensive history. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 7. Su m m a r y 102 A review of group duality intended for a physics audience [30] is being prepared for publication. General works written by mathematicians on duality or aspects of duality include [9], [52], [64, 65, 66 ], and [154]. In his solution to the problem of explicitly constructing all of the finite-dimensional irreps of general linear groups, Schur discovered what is now called Schur-Weyl duality, which he reported in his 1901 doctoral dissertation [145]. In his book [158], originally pub lished in 1928, Weyl gave wide publicity [125] to Young’s work on the symmetric groups [167] (including coining the term Young tableau), extended the work of Schur (Weyl’s character formula), and applied Schur’s discovery in quantum mechanical contexts. Since those early seminal works, Schur-Weyl duality has been used to develop branch ing rules and tensor product decompositions for a wide range of groups of interest in the physics community. In a series of publications, D.E. Littlewood used Schur-Weyl duality and particularly Schur function techniques [163, 103,104] to advance the theory of group characters. His major work on character theory [97] includes references to earlier work. In more recent times, R.C. King, B.G. Wybourne and their co-workers have continued to develop the character theory of Lie groups, and their work illustrates the power of Schur function techniques. The publications 1 [78], [76], [77], [10], and [11] provide entry points into their work on the character theory of compact Lie groups. Schur function techniques have also been used to extend the application of Schur-Weyl duality to the character the ory of noncompact Lie groups [153, 80, 55, 81, 82, 83], Hecke algebras [79, 161], and supersymmetry [12, 25, 26, 27]. Haase and Butler have used Schur-Weyl duality to derive relationships among coupling coefficients of symmetric and unitary groups [57, 58]. Also noteworthy is the work of D’Hoker [36] and Koike and Terada [85], where Young diagram techniques for unitary groups are systematically extended to the other types of classical groups, and the work of Brauer [13] in extending Schur-Weyl duality to a case applicable to quantum groups and knots and links [9]. As for the history of unitary-unitary duality, it seems to be obscure. The following passage is quoted from page 17 of Howe’s [ 66 ]. (What we call bosonic unitary-unitary duality, Howe calls (GLn, GLm)~duality, the basic theorem of which is his Theorem 2.1.2.) “I do not know the earliest reference for Theorem 2.1.2. W ith our current understanding of the symmetric functions known as Schur functions as char acters of the representations of GLn, we can see this result as implicit in a combinatorial identity known to Cauchy [103]. However, the ingredients necessary for the interpretation of the identity as Theorem 2.1.2 were not available until Schur’s thesis [145] in 1901, and I do not know when such an interpretation was first made, or whether it was the first proof of Theorem 2.1.2. As far as I can see from reading Weyl’s book [157], he did not under :Work done on branching rules until the mid-1960s is summarized by Whippman in [159]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 7. Su m m a r y 103 stand Cauchy’s identity in this fashion, though he certainly knew both it and the character formula for GLn. It is hard to imagine him not writing a very different book if he had understood this result. In any case, Theorem 2.1.2 seems not to have been recognized as common knowledge even in recent times, and to have been rediscovered by numerous authors, with various proofs, see for example [24], [54], [65], etc .”2 In the physics literature, fermionic unitary-unitary duality is implicit in the 1937 work of Wigner [160], and seems to have been discussed explicitly first by Moshinsky [106]. The first explicit studies of bosonic unitary-unitary duality in general that I am aware of were undertaken by Moshinsky [105] and Louck [102]. Earlier, Bargmann and Moshinsky studied the special case of N particles in 3-dimensional space [3, 4]. A recent application of unitary-unitary and symplectic-orthogonal dualities in optics and quantum interferometry is [138]. Other recent applications of unitary-unitary duality involve the calculation of SU(3) Clebsch-Gordan coefficients [135] and the calculation of SU(3) Wigner functions [138]. The use of symplectic groups in physics was brought to the attention of the physics community by Racah [120]. USp-USp duality was discovered by Helmers [63], who coined the term “group complementarity.” He built on the work of Flowers [45] and others in the classification of nuclear shell model states in j j coupling. Independently, Kerman introduced the concept of a quasispin group in [75]. The concept of USp-USp duality was summarized in works directed to physicists by Parikh [114] and Lipkin [93]. Subsequently, USp-USp duality has been widely applied to situations involving pairing, both in nuclear systems and superconductivity. Recent applications are described in [33], [155], [150], [101], [42], [112], [137], and [124]. Moshinsky and Quesne extended the work of Helmers and Kerman by construct ing examples of dual pairs of other groups [107, 108], leading to their observation of symplectic-orthogonal duality in [109]. A proof of symplectic-orthogonal duality in this context was obtained by Chacon in [ 22]. Symplectic-orthogonal duality is also im plicit in the work of Lohe and Hurst [100]. However, the history of symplectic-orthogonal duality in the mathematics literature predates these works. The oscillator representations of the noncompact symplectic groups were constructed and discussed by Segal [147], Shale [148], and Weil [156]. Further historical remarks on subsequent developments can be found in [50, 51, 67]. A notable subsequent paper is that of Kashiwara and Vergne [74], in which the authors state a number of theorems about symplectic-orthogonal duality. Howe generalized the concept of symplectic-orthogonal duality with his definition and investigation of reductive dual pairs of subgroups of a noncompact symplectic group [65] .3 There has been a very large body of work published in the mathematics literature on 2Howe also refers to another work here, but the citation is not present in his bibliography. 3Although this paper was published in 1989, the editor’s note begins, “This paper was written in 1976 and has been widely circulated as a preprint.” Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 7. Su m m a r y 104 what is often known as Howe duality, a summary of which is beyond the scope of this thesis. Entry points into this vast literature include [ 66 ], [52], and [92]. Howe duality has been extended to the realm of quantum groups [53], and has also been applied to gauge thoeries [144] and to the quantization of constrained systems [87]. Schur-Weyl duality has been extended to the exceptional Lie group G 2 in [69], and Howe duality has been extended to exceptional Lie algebras in [ 68 ]. For example, in [ 68 ] it is proved that (all algebras over C) • g 2 x so (3) is a dual reductive pair w ithin f 4 • g 2 x si (3) is a dual reductive pair within e 6 • g 2 x sp(3) is a dual reductive pair within e 7 • g 2 x f4 is a dual reductive pair within e 8 Further duality relationships involving exceptional groups are stated in [38] and [92]. Rowe, Wybourne, and Butler [141] made use of symplectic-orthogonal duality in ex tending the character theory of compact groups to noncompact symplectic groups. They showed that the characters of /c-fold tensor powers of the oscillator representations of noncompact symplectic groups can be expressed in terms of well-defined infinite series of Schur functions, in effect obtaining the branching rules Sp(n, R) J, U(n). Symplectic- orthogonal duality has been used for determining branching rules [136], for resolving U(n) D O(n) missing label problems [89, 90, 91], [34, 116, 117, 118], and for determining the matrix elements of operators [128, 132]. Such applications of symplectic-orthogonal duality were motivated by, and are essential to the implementation of, the nuclear sym plectic model. 7.2 Further research in group duality Group duality is a powerful tool in physical applications that continues to be widely used. Besides the im plicit use of Schur-Weyl duality in bread-and-butter applications of symmetry analysis to many-body physics, new applications of group duality continue to appear in research papers. In addition to the many examples already listed in the previous section, a very recent application of duality to error correction in quantum computing is [73]. This underscores the fact that although our discussion of Schur-Weyl duality has centred on its applications in the construction of multi-particle wave functions with particular permutation symmetries, it is applicable in any situation where one wishes to classify the symmetry properties of the combination of multiple systems. In considering future work in group duality, a number of interesting problems seem worthy of consideration. Practically speaking, whenever a situation of physical interest Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 7. Su m m a r y 105 involves a chain of groups, one can search for a dual chain of groups that may facilitate calculations. In addition to such practical applications, one may endeavour to develop a deeper conceptual understanding of group duality. For instance, in many of the examples considered in this thesis, one of the groups in a pair of dual groups is a dynamical group and the other is a symmetry group for the system of interest. One wonders how general this situation is, and whether there are criteria that specify in which circumstances one obtains this physical interpretation of dual groups. Such criteria may improve one’s understanding of systems that are described using dual pairs of groups. A related issue is the geometry of group duality. There is a dual-group structure un derlying the method of separation of variables. Orthogonal coordinate systems facilitate the solution of problems, but dual group structures are more general, and are helpful even when global orthogonal coordinates are not present. For example, in the symplectic nuclear model discussed in the previous chapter, symplectic-orthogonal duality allows one to separate the dynamics into collective and intrinsic parts. The states that carry each Sp(3, R) irrep correspond to a set of collective states, with different intrinsic degrees of freedom being given by the states that carry irreps of the (dual) 0 (N) group. The geometric analogue of this is that the group manifold for Sp(3A, R) can be factored into collective and intrinsic submanifolds. A study of the geometry of group duality might shed light on the meaning of dynamical and symmetry groups, and the strategy of sepa rating dynamical motions into collective and intrinsic components. Investigations of the group theory behind the method of separation of variables, using the idea of contractions of groups and algebras but not group duality, can be found in [70, 71]. Continuing with the theme of geometric structures that may be related to group duality, the possible connections between group duality and both the orbit method and geometric quantization would make an intriguing research project. The orbit method associates the set of equivalence classes of unitary irreps of a group w ith the set of coadjoint orbits, that is the set of orbits of the action of the group on the dual of its Lie algebra. (One problem with the orbit method is that this association is not always 1- 1 . Adams has studied the conditions for which the 1-1 correspondence of irreps in a reductive dual pair is reflected by a 1-1 correspondence in the associated coadjoint orbits [1, 2].) In geometric quantization, a coadjoint orbit (which is a classical phase space) is associated with a Hilbert space that carries a single irrep of a relevant group, thereby relating a classical model with a corresponding quantal model. Also important in this complex of ideas is the theory of induced representations. Vector coherent states theory at once unifies much of this discussion and also extends the theory of geometric quantization by allowing for the quantization of intrinsic degrees of freedom. For more information on all of these topics, one can consult [133, 6 , 7, 8] and [5]. Another interesting paper exploring the relationship between group duality and geo metric structures is [144], where the structure of the orbit space of the action of a gauge group on a space of gauge potentials is studied using Howe duality. The relationship between classical mechanics and quantum mechanics in the theory of geometric quanti Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 7. Su m m a r y 106 zation is reminiscent of Howe’s formulation of group duality within symplectic groups. Howe first makes a construction on polynomials, and then extends the construction to the oscillator representation [64, 65]; this is very much in the same spirit as geometric quantization and warrants further study. Finally, the whole concept of group duality does not seem to be well-understood, and one anticipates that further research w ill clarify some of its currently mysterious aspects. For instance, one would like to have criteria that could be applied to a pair of groups (and a Hilbert space on which they both act) that would specify definitively whether the groups are dual. That the group actions commute is essential, and that the groups be mutual centralizers also seems essential, but that these two conditions are not sufficient to guarantee duality is shown a few paragraphs hence. (The examples dealt with in this thesis do not suffer from such complications, because of the way that creation operators are used to construct the spaces that carry the representations.) The fact that there are several different definitions in the literature is a sign that a better definition of group duality may yet be formulated. For instance, we have the definition of Moshinsky and Quesne [108], which is sim ilar to the one we use, although more restrictive: Consider the direct product of groups G\ and G2, which are subgroups of a larger group G. Then Moshinsky and Quesne define G\ and G2 to be complementary, on a space carrying a definite irrep of G, if there is a one-to-one correspondence between all the irreps of G\ and G 2 that are contained in the specified irrep of G. In the examples that Moshinsky and Quesne cite, they begin with groups G and G\, then construct G 2 as the group of all operators within G that commutes with Gi (that is, G 2 is the centralizer of G1 w ithin G), and then go on to show that the irreps of G\ and G 2 are in one-to-one correspondence within the Hilbert spaces of interest. In Howe’s definition [64, 65], G is a symplectic group, and the pair of subgroups G 1 and G 2 are defined to be dual provided that they are mutual centralizers within G. Using a H ilbert space H that carries the oscillator representation of G, he was able to prove that if G i and G 2 are dual in their actions on H, and one of G\ and G 2 is compact, then H decomposes into a discrete direct m ultiplicity-free sum of irreps of G\ x G2, such that the irreps of G i and G 2 that appear in the decomposition are in one-to-one correspondence. One can appreciate how special the oscillator representations of symplectic groups are by considering the following example. Any two groups G\ and G 2 w ith triv ia l centres and whose actions commute are mutual centralizers within the direct product group Gi x G 2 (i.e., in their actions on a Hilbert space that carries a representation of the direct product group). Yet the irreps of Gi and G 2 w ill not be in one-to-one correspondence, unless the situation is trivia l. So even if the groups are mutual centralizers, there must be some other condition present in cases for which G\ and G 2 are actually dual. The question is, what is this other condition that makes irreps of dual groups interlaced, whereas those of Gi and G 2 w ith in G\ x G 2 are unrelated? The answer may lie in some special property of the containing group, or of the Hilbert space that carries the duality, and the identification of such a property would be of great interest. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h a p t e r 7. Su m m a r y 107 Finally, Klink and Ton-That [84, 154] define duality as follows. Let R be a represen tation of G\ and let L be a representation of G 2, both carried by a H ilbert space H, such that the actions of the two groups on H commute. Assume that the representations R and L can each be decomposed as the direct sum of irreps. Then R and L are said to be dual representations if the decomposition of R into irreps determines that of L and vice versa. Thus, Klink and Ton-That speak of dual representations rather than dual groups. This definition is general enough that it includes all of the examples discussed in this thesis (and a great number of dualities that are discussed in the literature). However, since the definition places no conditions on the groups G 1 and G2 (besides that they commute, so that their direct product can be formed) it is unlikely that it will lead to an understanding of the special properties of certain groups and the Hilbert spaces upon which they act that brings about the phenomenon of duality. In any case, the question of determining simple criteria by which one can determine whether two groups are dual is s till open. Perhaps, as has been the case a number of times in history, once a more fertile definition of group duality is discovered, this issue, and some of the others mentioned in this section w ill be resolved. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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[154] Tuong Ton-That, Dual representations and invariant theory, in Representation The ory and Harmonic Analysis , edited by Tuong Ton-That, Contemporary Mathemat ics, 191, American Mathematical Society, Providence, RI, 1995. [155] P. Van Isacker, Dynamical symmetries in the structure of nuclei, Rep. Prog. Phys. 62 (1999) 1661-1717. [156] A. Weil, Sur certains groupes d’operateurs unitaires, Acta Math. I l l (1964) 193- 211. [157] Hermann Weyl, The Classical Groups, Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1946. [158] Hermann Weyl, The Theory of Groups and Quantum Mechanics , Dover, New York, 1950. [159] M.L. Whippman, Branching rules for simple Lie groups, J. Math. Phys. 6 (1965) 1534-1539. [160] E. Wigner, On the consequences of the symmetry of the nuclear Hamiltonian on the spectroscopy of nuclei, Phys. Rev. 51 (1937) 106-119. [161] B.G. Wybourne, Fun with Hecke algebras H n(q) of type i, in Symmetry and Structural Properties of Condensed Matter, edited by W. Florek, T. Lulek, and M. Mucha, World Scientific, Singapore, 1991. [162] B.G. Wybourne, The representation space of the nuclear symplectic Sp( 6 , R) shell model J. Phys. A 25 (1992) 4389-4398. [163] B.G. Wybourne, Symmetric Functions and Their Application to Problems in Physics, in Symmetry and Structural Properties of Condensed Matter, edited by W. Florek, D. Lipinski, and T. Lulek, World Scientific, Singapore, 1993. [164] B.G. Wybourne, Plethysm in Physics and Chemistry Applications, in Symmetry and Structural Properties of Condensed Matter, edited by T. Lulek, W. Florek, and B. Lulek, World Scientific, Singapore, 1997. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B ibliography 120 [165] Brian G. Wybourne, Symmetry Principles and Atomic Spectroscopy , Wiley, New York, 1970. [166] M. Yang, An algorithm for computing plethysm coefficients, Discrete Math. 180 (1998) 391-402. [167] Alfred Young, The Collected Papers of Alfred Young , edited by G. de B. Robinson, University of Toronto Press, Toronto, 1977. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A Plethysms of Schur functions and the shell model M.J. Carvalho and S. D’Agostino J. Phys. A: Math. Gen. 34 (2001) 1375-1392 Received 17 July 2000, in final form 2 January 2001 A bstract We present a method for evaluating plethysms of Schur functions that is conceptually simpler than existing methods. Moreover the algorithm can be easily implemented with an algebraic computer language. Plethysms of sums, differences, and products of S-functions are dealt w ith in exactly the same manner as plethysms of simple S-functions. Sums and differences of S-functions are of importance for the description of multi-shell configura tions in the shell model. The number of variables in which the S-functions are expressed can be specified in advance, significantly sim plifying the cal culations in typical applications to many-body problems. The method relies on an algorithm that we have developed for the product of monomial sym metric functions. We present a new way of calculating the Kostka numbers (using Gel’fand patterns) and give, as well, a new formula for the Littlewood- Richardson coefficients. A.l Introduction In group theoretical models of many-particle systems, of particular importance is the construction of iV-body states that simultaneously belong to definite irreducible repre sentations (irreps) of both the model’s relevant group G and the symmetric group Sn - For instance, wave functions that describe N fermions must be totally antisymmetric with respect to any permutation of the N fermions; hence to construct states of well- defined permutation symmetry one considers as a subgroup of the chain of groups 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 122 relevant to the model. Mathematically this requires the decomposition of tensor power representations of G into representations that have a particular symmetry type w ith re spect to particle permutations; this operation on the group characters is known as the symmetrized power or plethysm. The groups of interest in many-body physics are the classical compact Lie groups (general linear, special linear, unitary, special unitary, orthogonal, special orthogonal, and unitary symplectic), as well as the non-compact symplectic Lie groups, and of course the symmetric groups. Schur functions (S-functions for short) are characters of the unitary irreps of unitary groups (and the characters of finite-dimensional irreps of general linear groups) [1]. The plethysm of S-functions was introduced by Littlewood [2], who developed a number of useful techniques for its calculation [1], Plethysm of S-functions applies directly to prob lems involving irreps of unitary groups. However, it is well known [3] that the characters of the unitary irreps of all the other aforementioned Lie groups are expressible in terms of S-functions, and vice versa. To calculate a plethysm for any such group G, then, one expands the characters of G in terms of S-functions, determines the plethysm of the S-functions, and then expresses the resulting series of S-functions back in terms of the characters of G. This procedure is straightforward in principle for compact groups, although the calculations may be difficult in practice. For non-compact groups, where the characters are infinite series of S-functions, the difficulties are even greater. Methods for evaluating plethysms of, for example, the fundamental irreps of non-compact sym plectic groups have been given in the literature [4-8]. These make use of the generating functions of S-function series and enable one, in principle, to obtain the fu ll expansion of such plethysms, though in practice one has to truncate them at a prescribed cutoff. In any case, the plethysm of S-functions is of fundamental importance for the calculation of plethysms of irreps for any of the aforementioned groups. The practical difficulties in calculating plethysms have stimulated a continuing search for algorithms that are both simple and efficient. Some breakthroughs include a notable paper by Butler and King [9], in which they obtain recurrence relations for plethysms; the algorithm for plethysm of Chen et al [10]; and the recently published method of Yang [11] for evaluating the coefficient of a single S-function in the expansion of a plethysm of two S-functions. Also computer codes exist to evaluate S-function plethysms; for example “SCHUR” . 1 If the plethysm of simple S-functions is arduous in practice additional difficulties arise when one needs to calculate symmetrized tensor powers of a sum, difference or product of S-functions, which we call here for brevity compound S-functions. Since plethysm is not distributive on the left, the evaluation of such plethysms has up to now relied 1SCHUR™ An interactive program for calculating properties of Lie groups and symmetric functions distributed by Steven M Christensen and Associates, Inc., PO Box 16175, Chapel Hill, NC 27516 USA. E-mail [email protected], webpage http://smc.vnet/Christensen.html Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 123 upon complicated manipulations of S-functions as expressed in equations (A.44-A.46) of Section V. Physical situations where plethysms of compound S-functions are required include: i) Identification of nuclear shell model states that span irreps of appropriate unitary groups and have a particular permutation symmetry for a system of particles in a multi-shell configuration. ii) Classification of many-particle states, of well-defined permutation symmetry, by rep resentation characters that are not simple S-functions; for example, by characters of irreps of the orthogonal or unitary symplectic groups. The objective of this paper is to outline a method for evaluating plethysms of S- functions that is conceptually simpler than existing methods. The method is simpler because it only requires a straightforward algorithm for multiplying monomial symmetric functions and no other complicated rules intervene. Moreover the algorithm can be easily implemented with an algebraic computer language such as Maple. A major strong point of the method is that plethysms of compound S-functions are dealt with in exactly the same way as plethysms of simple S-functions. The number of variables in which the S-functions are expressed can be specified in advance, significantly sim plifying the calculations in typical applications to many-body problems. In essence, the method described here consists of i) converting the S-functions to monomial symmetric functions, ii) calculating the plethysm of monomial symmetric function series, where distributivity on the left is valid, and Hi) converting the result back to S-functions. The feasibility of this method relies on an algorithm (presented in Section IV) for the product of monomial symmetric functions, which to the best of our knowledge has not previously appeared in the literature. The structure of the paper is as follows. In Section II we present the essential facts about partitions and symmetric functions and establish the notation used. In Section III we review the interrelations among monomial symmetric functions, power-sum symmetric functions and S-functions, and give a novel method for determining the transition ma trices for S-functions in terms of monomial symmetric functions using Gel’fand patterns. In Section IV we review the outer product of S-functions and its physical interpretation. We also give an algorithm (developed in detail in Appendix B) for the product of two monomial symmetric functions. This algorithm is used in section IV to resolve the outer product of two S-functions, incidentally obtaining a new formula for the Littlewood- Richardson coefficients, and in Section V to obtain an algorithm for the resolution of the plethysm of two S-functions. The physical interpretation of plethysms of S-functions is discussed in Section V. Section V I contains an example illustrating the application of the new algorithm to the plethysm of compound S-functions, and concluding remarks. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 124 A.2 Partitions and symmetric functions Polynomials in r independent indeterminates x \, x2,..., x r that remain invariant with respect to arbitrary permutations of the indices of the indeterminates are known as symmetric functions. Symmetric functions of degree n are labelled by partitions of n. For further details on symmetric functions and partitions one can consult, for example, the books by Littlewood [1] or Wybourne [3]. A comprehensive source for information on symmetric functions and partitions is Macdonald’s book [12]. In this section we shall provide a brief summary for the reader’s convenience. A partition A = (Ai, A2,..., A*) of the positive integer n is a sequence of positive integers Aj (the parts of the partition) for which |A| = )Cj=i A* = n where |A| denotes the weight of the partition A. The notation A h n indicates that A is a partition of n. The partition is said to be standard provided that Ai > A 2 > • • • > A*,. The number of non-zero parts of a partition is called the length of the partition, l( A). Exponents, called multiplicities, are commonly used to sim plify the notation for a partition. For example, (4422211111) can be w ritten as (422315), where = 5, v2 = 3, vz = 0, and v± — 2. Partitions are easily visualized by using Young diagrams. Specifically, the Young diagram associated with a partition A = (Ai, A2 ,..., A*,) consists of k rows of boxes and has Aj boxes in its *-th row. The main diagonal of a Young diagram consists of the first box in the first row, the second box in the second row, and so on. By reflecting the Young diagram for A in its main diagonal one obtains the Young diagram for its conjugate partition A'. For example, the Young diagrams associated with the partition (3212) and its conjugate (421) are □ (A .l) □ □ In what follows we’ll sometimes need to add zeros to the end of a partition in order to have a prescribed number of parts. The resulting partition is equivalent to the original one. For example the partitions (21), (210), (2100), and so on, are all equivalent. In this paper we are only concerned with three types of symmetric functions, the monomial symmetric functions m\, the power-sum symmetric functions pj, and the S- functions s\. The set of all symmetric functions labelled by A b n form a vector space for which the sets {m \ | A h n }, {p^ | A b n} and {s* | A b n } are bases. Consider a fixed number r of variables (indeterminates) x \, x 2,..., xr \ then, the S- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 125 function denoted by either 2 S\ or {A}, is defined as x: «A (A.2) \X'k—t I where s and t index rows and columns respectively of the r x r determinants. For example, in terms of three (r = 3) indeterminates, the S-function labelled by A = ( 2 1 ) (recall that (2 1 ) = (2 1 0 )) is given by 2 + 2 X 1 z ? + 0 2 + 2 - 1 + 1 - 0 + 0 X2 + 2 + 2 2 + 2 r l+ l —0 + 0 X3 X3 X3 5 (2 1 0 ) x\ X5 - 0 x \ x \ x 2 r o xi x \ x 3 or 5 (2 1 ) = xlx2x3 + X1X2X3 + X1X2X1 + X1X2X3 + X\X2X\ + X\X2X\ + 2 x {x \x l . (A.3) One of the facts that makes S-functions so useful is that these formulas are essentially independent of the number of variables; the exception is that if there are insufficient variables, then some S-functions are identically zero. To be precise, an S-function w ith k parts in r variables is identically zero if r < k. Equation (A.2) follows from the Weyl character formula applied to unitary groups, so the characters of unitary irreps of U (r) are in one-to-one correspondence with S-functions in r variables. Also note that an S- function s\ is said to be standard if the partition A that labels it is a standard partition. However non-standard S-functions can also be defined. A non-standard S-function is either zero (if A*_i = Aj — 1 ) or can be converted to a standard one by using the well known S-function modification rule [3] ®(Ai,...,A,-i,Aj,...,Aj.) l,Aj-i+l,...,Afc) ? (A.4) which is a consequence of the properties of determinants, more precisely the determinant in the numerator of equation (A.2). In terms of the r indeterminates, the monomial symmetric function m \ (m-function for short) is defined as /y.^1 Ait U/1 «A>o •X (k < r) , (A.5) 2 Both notations are widely used in the literature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 126 where the label A stands for the partition A = (Ai, A2,.. ■, A*), and the sum includes all distinct terms obtained by all possible permutations of the subscripts i of the indeter minates Xi. The number of terms in the sum (which henceforth we call the dimension dim(m>) of the m-function), is given by f\ dim(mA) = vv r , (A.6 ) l l j Pi- where p* for i > 1, are simply the multiplicities of the parts of A and p,0 = r — k. Thus, for example, for r = 3 and A h 3 we have the three m-functions = x f + x \ + x%, 2 1 ) = x \x 2 + x ix \ + x\x$ + X \x l + x\xz + x 2x \, and m(m) = X ix 2x$, w ith dimensions dim (ra(3 oo)) = 3, dim (m ( 2 io)) = 6 and dim (m (m )) = 1. The power-sum symmetric function pj is the sum of the j- th powers of the r indeter minates: r Pj = J2 Xi =m 0 ) ’ (A '7) t=l For example, for r = 4 and j = 3, p 3 = x \ + x% + ref + x \. In addition, the p-function p\, where A = (Ai, A2,..., A*), is defined as P a = P a 1P a 2 ■ ■ ■ P a , ■ (A -8) A.3 Transitions Among Symmetric Functions In order to implement the plethysm procedure in Section V we need to calculate elements of the transition matrices that interrelate the above-mentioned bases for symmetric func tions. Note that in what follows we assume that the related symmetric functions are expressed in the same variables. A.3.1 Expansion of S-functions in terms of p-functions The remarkable relationship between the unitary and symmetric groups, known as the Schur-Weyl duality, leads to a direct relationship between a character x x of the symmetric group and the corresponding character s\ for a unitary group. For A h n one has = £ 7 1Pp ’ (A -9) 0 P P where p = (pi, p2, . ..) labels the conjugacy classes of Sn, pp = P/h are p-functions, x P are the components of the character x X °f the irrep A of Sn, and zp is given [ 1 2 ] by zp = n1 ! l i/V 2 !22'2 • • • un\rUn . (A.10) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 127 A.3.2 Expansion of S-functions in terms of m-functions The S-function s\ is expressed in terms of m-functions as follows s\ = K >* m a* ’ (A-ll) l/*l=|A| where fj, is a standard partition of |A| w ith / / 1 not exceeding Ai. The coefficients Kxp, which are either positive integers or zero, are known as the Kostka numbers [12]. Several methods are given in the literature for the determination of the Kostka numbers Kx». Typically they are calculated by combinatorial means involving Young diagrams. The simplest is to list the standard numberings of the relevant Young tableaux with integers in the range 1 ,... ,n such that the numbers (/jj, /i2,...) of occurrences of the integers ( l , 2 ,...,n) are such that ni+\ < /^; e.g., for the f/(3) S-function S(21) the list contains the numbered tableaux 1 1 1 2 1 3 (A.12) 2 3 2 corresponding to g, — (210), (111), and (111), respectively. This gives S(2 i) = ^ ( 2 1 ) + 2 m (m ). (A.13) In Appendix A we explain the physical meaning of Kostka numbers in terms of represen tation theory. However, for computational purposes, it is more efficient to identify and enumerate the partitions g in (A.ll) by means of Gel’fand patterns. For a given S-function (in r indeterminates), labelled by A = (Ai, A2,..., A*), the m- functions appearing on the r.h.s. of (A .ll) can be identified by means of triangular arrays of integers (the Gel’fand patterns) [13]. g ii g 12 <7i3 5 l r - l 5 lr 521 522 523 52r—1 531 532 53r—2 (A.14) grr whose entries are subject to the following conditions Ai for i < k 5 li 0 for k < i < r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 128 9k—l,i ^ 9ki ^ 9k-l,i+l ■ (A.15) The parts of each partition p are given by differences of the sums of the entries in two successive rows of the array r—i+ 1 1=1 9 r ~ Qrr ■ (A.16) It is clear that for a partition A there are many compatible patterns. In fact, there are as many as 52^ dirr^m^). In other words, each possible Gel’fand pattern gives one term of each m-function that comprises s\ ; equivalently, the Gel’fand patterns are in one-to-one correspondence with the semistandard Young tableaux of shape A. However one does not need to construct all of the Gel’fand patterns in order to identify the relevant m- functions. It is enough to recognize the distinct patterns that give rise to leading terms, i.e. those for which Hi > n i + This selection can be efficiently implemented by imposing on the entries of these patterns the extra condition r r—i—1 (A.17) Consider, for example, the S-function S( 2 i) in three indeterminates. The triangular pat terns that satisfy the required conditions are 2 1 0 2 1 0 2 1 0 1 0 2 0 1 1 (A.18) 0 1 1 Thus, 5 (2 1 ) = m (2 i) + 2m(1U). A.3.3 Expansion of m-functions in terms of S-functions An m-function m \ is expressed in terms of S-functions as (A.19) where the coefficients K ^ , which can be either positive or negative integers, are the inverse Kostka numbers. The number of terms in the sum (A. 19) cannot exceed the dimension of and depends on the number of indeterminates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 129 It may appear that the coefficients can be obtained trivially by inverting the appropriate Kostka matrix. However that method is rather uneconomical in the context of plethysm calculations because one typically requires a particular row of the inverse of a Kostka m atrix that does not necessarily need to be constructed (since the S-functions in the final result have higher degree than the original S-function factors). A practical, easily programmable procedure for determining the inverse Kostka numbers requires only two straightforward steps conveyed by the following formula 3 • (A-2°) A » where the first sum runs over all distinct partitions A generated from A by permuting its parts in all possible ways. Clearly, only one of the partitions A is standard. Application of the modification rules to the non-standard S-functions leads to sa and the sought-after inverse Kostka numbers. As an example, consider the m-function m ^ i) in three indeterminates. We have that 7^(321) = S(32l) + 5(312) + 5(231) + 5(213) + 5(123) + 5(132) • (A.21) Since, by the m odification rules, 5(312) = 5(231) = 5(123) = 0 5(213) — — 5(222) 5(132) = —5(222) j (A .22 ) then the resulting S-function expansion (in three indeterminates) is wi(321) = 5(321) — 2S(222) • (A .23) A.4 Products of symmetric functions A.4.1 Physical interpretation of S-function products Each operation involving S-functions corresponds to an operation on unitary represen tations of G L (n) or its subgroups [16]. In particular, outer products of S-functions correspond to tensor products of unitary irreps of U (n) or G L {n) . 4 To see that outer 3Note that this procedure arises from multiplying an m-function by a Vandermonde determinant and rearranging the result as discussed in [1, 14]. We would like to thank a referee for pointing out that this procedure can be traced back at least as far as Muir in 1882; see [15, pp 150-151]. 4The results obtained are independent of n, and are therefore particularly powerful. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 130 products of S-functions have fundamental physical importance for descriptions of many- particle states consider the following. Suppose that a state of Np protons is specified by s\ and a state of N n neutrons is specified by sp, so that A b Np and p fr N n. Then a state of the combined system of Np + Nn particles is specified by an S-function sa that occurs in the expansion of the outer product = ^ ^ , (A.24) a where each partition a is a partition of Np + N n. Note that all of the S-functions s^, sp and sa are irreps of the same unitary group U(n). Another S-function operation that is of importance in physical applications is plethysm or symmetrized power. Consider single particle states labelled by corresponding to an irrep of U(n); then states of N identical particles w ith permutation symmetry [v\, v h N, are labelled by an S-function sa occurring in the expansion of the outer plethysm 5 ®A ® Su = ^ ^ AXvaSp ■ (A.25) a In equation (A.25), v labels an irrep of Sn and sa are irreps of U (n). More details about this operation and the new algorithm to evaluate it are deferred until Section A.5. A.4.2 Product of m-functions The product of m-functions is a simple product of polynomials. In order to develop an efficient algorithm for this product without having to work out all the terms explicitly, let’s first define the addition of two partitions (oq, a2,...) and (/?i, /32, • • •) as being the partition whose parts are (ati + fa, cc2 + An • • •)• ^ is necessary that the partitions have an equal number of parts; if they don’t, then one increases the number of parts of the shortest one by adding enough zeros at the end. The product of two simple m-functions is then defined as mam,p = ^ 2 > (A.26) where the partitions 7 result from adding to a all distinct partitions /3 obtained by permuting in all possible ways the parts of /?. Note that one reorders the parts of the resulting partitions 7 to make them standard. The multiplicity / 7 of a resulting m- function m 7 is given by / % , (A.27) dim (m 7) 5Since in this paper we only refer to outer plethysm we shall drop the adjective “outer” from here on. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 131 where n 7 is the number of times the same partition 7 appears in the process of adding partitions referred to above. One can determine n 7 as follows. Write a in the form (k^h(k — 1 ) ^ - 1 ... where the Hi are the multiplicities defined as in equation (A.6 ). Then H7 = P k P k -i' ' • P i Pq (A.28) where is the number of distinct permutations of the first /i* parts of the partition f3 (not necessarily in standard form), Pk-\ is the number of distinct permutations of the next jj,k- 1 parts of /3, and so forth, where 7 = a + j3. Clearly, all m-functions on the left or right of equation (A.26) are functions of the same r indeterminates. If all the possible partitions 7 are to appear in the expansion (A.26) (i.e. the expansion is complete) then r should be set equal to the sum of the lengths of a and /3. However, if the product (in equation A.26) is part of a calculation involving irreps of the unitary group U (n ) then one should set r = n, since S-functions w ith more that n parts are identically zero in U(n). As an example consider the product m( 32)1^ ( 11) and choose r = 4 indeterminates so that the resulting expansion is complete. The addition of a = (3200) = (32) to the list ( 1 1 0 0 ), (1 0 1 0 ), (1 0 0 1 ), (0 1 1 0 ), (0 1 0 1 ), (0 0 1 1 ) (i.e. to the partitions generated from (1 1 0 0 ) = ( 1 1 ) by perm uting its parts in all possible but distinct ways) gives: (4300), (4210), (4201), (3301), (3310), (3211) . (A.29) W ith the values given in table 1 , where use was made of (A. 6 ), we get the final result 171(3200)171(1100) = 171(4300) + 171(4210) + 2l?2(33io) + 112(3211) - 7 n 7 dim (m 7) t 7 (4300) 1 60 1 (3310) 2 60 2 (4210) 2 1 2 0 1 (3211) 1 60 1 Table 1. Multiplicities of the m-functions in the expansion of the product m( 3 2 )m(n) in r = 4 indeterminates. Note that if we had chosen r = 3 instead (as in the context of U(3)), then the calculation would have involved only partitions in no more than three parts; i.e., a = (320) and /3 = (110). In this case, the result obtained is, U1(320)1H(110) = 111(430) + Hl(421) + 2 771(331) > (A.30) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 132 which, as expected, is the same as (A.30) except for the absence of the partition with 4 non-zero parts. The algorithm for the m ultiplication of two m-functions (cf. eq. (A.26)) can be easily generalized to the multiplication of series of m-functions. The result is (for more details see Appendix B) ) E c^m«i2 •" = E T'n»-pm ™ ~ p ’ (A-31) V *1 / V 12 / \ ip J 712...P where the coefficients V , ~ E l '^>7l2...P-l.-.««^7l2...cl7l2...n-lj.fal (A.32) j^P can be found recursively. For example ^712 ~ E / ^7»1 ,i2^7i2{7»i,i2} ’ (A.33) Jl,l2 where <57l2{7il i4 } = 1 if the partition 7 12 appears in the set { 7 ,,^} and zero otherwise. According to equation (A .27), dim (m a.i ) !dim(m7jii2) ^ 7 1 2 3 ^ (A.35) his w ith dim(m~,,..) ^ , < 3 = E T ^ 2 ^ n7 i2J, i3 dim(m71J ’ (A’36) M3 and so on. A.4.3 The Littlewood-Richardson coefficients The standard method for evaluating the r.h.s. of (A.24) is the well-known Littlewood- Richardson rule [17], which is a set of directives to be applied to the corresponding Young diagrams. Although application of these rules is feasible in simple cases, their execution for partitions of large numbers is rather complex. The resolution, given below, of the outer product of two S-functions via the product of m-functions is simpler and much more Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 133 amenable to automatic computation. Using the algorithm for the product of m-functions, one has for the outer product of two S-functions, SjuSj/ K upi m p '^ (A.37) y ; T 7m7 (A.38) 7 (a . 3 9 ) O 7 where T 7 = T 7l2; cf. (A .33). Note that we have now obtained a new formula for the Littlewood-Richardson coef ficients, F = V ' 1 jiva Z —/ K~lT 717 7 = E ^ ' E ^ w 7 ij = E * £ E ■ (A-4°) Note also that this formula enables one to calculate the coefficient of a single S-function in the expansion of the outer product without having to construct the whole expansion. A.5 Plethysm of S-functions The plethysm of two S-functions sx and sM, of weights |A| = n and \/j,\ = k respectively, Sa © S/4 = ^ ^ Aa[hjSu (A.41) c h n k gives a sum of S-functions, all of weight n k , with non-negative integer coefficients AX/ia. Plethysm is a symmetrized power of S-functions, i.e. the outer product of k copies of sx can be decomposed into a sum of sets of terms, where the S-functions in each set have permutation symmetry (/i): 8\ 8\ y - s K = ^ 2 / ' ‘sa © S/4 , (A.42) k copies M where the sum on the right side extends over all partitions of k, and is equal to the dimension of the S k irrep labelled by /a . In more physical terms, the states of a Ai-particle Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 134 system (where each particle is individually described by sx) of permutation symmetry [n] are described by the plethysm sx © sM. For example the simple (outer) product si sx sx has expansion Si Si Si = S3 + 2s2i + Sin . (A.43) If one regards Si as the character of a 17(3) irrep spanned by the wave functions of a single particle, then the product Si Si Si is the character of the tensor product of three copies of this irrep. This reducible representation is spanned by a set of three-particle wave functions. It is a direct sum of three irreps: an irrep with character s 3 spanned by wave functions that are fully symmetric w ith respect to particle exchange; an irrep w ith character sm spanned by fully antisymmetric wave functions; and two mixed symmetry irreps with character S2 1 . Note that a given S-function may appear in different symmetrized products, so S- functions do not characterize the symmetry classes of tensor products. In contrast with outer product, plethysm of S-functions is not commutative or distributive on the left over outer product, addition and subtraction. Current algorithms cannot directly handle plethysms of compound S-functions (i.e. linear combinations or outer products of S-functions). The difficulty is due to the fact that plethysm of S-functions is not distributive on the left and so use must be made of [3] (.A + B)@ sx = £ I V a ( A ® 5 p) ( B ® Si,) (A.44) ( A - B ) @ s x = ^(-I^'IV a (A © s y ) (B © v ) (A.45) H,v (.AB ) © sA = Va (A © Sp) (B © s„) , (A.46) where A and B stand for either S-functiolis, characters of classical groups or any linear combination of S-functions. In equations (A.44) and (A.45) the coefficients T ^ x are taken from = Y2x F ^x^x and s„» is the S-function labelled by the conjugate partition of v. In equation (A.46) the coefficients kIJuX are taken from the internal (inner) product s^o sv = Y2x ^ a sA) (cf. [3]). It is assumed in all three equations that the summations include the cases sM = s0 = 1 and sv = sXl and vice versa. If use of equations (A.44), (A.45) and (A.46) is already tedious when A and B are simple S-functions, it becomes even more so when they are compound S-functions. In the following we present a method that permits the evaluation of the plethysm of a compound S-function without having to resort to the labour entailed by using equations (A.44), (A.45) and (A.46). By expanding the S-functions on the left of the plethysm sign © (eg. cf. l.h.s. of equation (A.44)) in terms of m-functions and the S-function on the r.h.s. in terms of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 135 p-functions, the plethysm of a compound S-function (or of a simple one) is then reduced to the plethysm of a series of m-functions w ith p-functions. But since the plethysm of an m-function w ith a p-function is still an m-function, (A.47) where j.p, means that each part of p, is m ultiplied by j (that is, if p = ( p i,..., pk) then j.p = (jpi, ■ ■ ■, j^k)), then evaluation of the plethysm of S-functions (simple or compound) only involves, just as for the outer product, the multiplication of m-functions. A.5.1 An algorithm for the plethysm of S-functions Here is a detailed description for evaluating the plethysm (A.48) a First, the S-function sv is expanded, as usual, in terms of the p-functions pp = f j , Pp, (A.49) p and since plethysm of S-functions is distributive on the right, s \® s u = '52XpZp1s\© Y lpPi p i (A.50) Now, the S-function, s\, is expanded in terms of m-functions (A.51) J and by making use of the property nip © pj = rrij.p (A.52) one obtains (A.53) It remains to evaluate the product Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 136 which can be done using (A.32) to yield p=i(p) / \ II \^2K^m Pi.Qj = T7f2...pm7f2...p • (A-54) «= 1 V i / 7fa...p Thus, sx©sv = Y J Xvpzp 1 Z T 7f2...pw 7f2...p = E (A ‘55) 7i 2...p w ith T7 = E*P*P lT 7f2.../7{7f2...p} ’ (A'56) where the set { 7 } is the union of all sets of partitions { 7 ^..^} f°r all classes p, and the delta function ^ 7 {7 f2 } ensures that the coefficient T 7 has contributions from the individual coefficients T 7 />2 when a partition 7 is common to more than one set { 7 i 2...p}. Converting the m-functions back to S-functions we finally have ® s, = Z E T 7 K 7~* ^ • (A.57) < T 7 Thus, Aaw = Z T 7 ■ (A-58) 7 The plethysm of a sum, difference or outer product of S-functions clearly reduces to the evaluation of an equation formally identical to (A.53), since a sum of a series of m-functions is still a series of m-functions. For example, Kp ) / \ (Sa + S„) © s„ = E x p V I I E K\ 0tTTlpi,a "h 1 p i \ a / A.6 Illustrative example and Concluding Remarks To illustrate the method presented in this paper, let us consider the plethysm in U (3) (s(l) + S(2 ) + S(3 )) © S(2 1 ) 1 (A.59) relevant to the physical problem of finding the U (3) representations that can describe 3 particles placed with permutation symmetry (21) in any of the valence shells N = 1, N = 2, or N = 3 of the spherical harmonic oscillator. We shall outline first how the calculation is carried out using equation (A.44) (which we refer to as method I), and then discuss the evaluation of the same plethysm by means Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 137 of the procedure advocated in this paper (method II) and implemented with Maple. Of course the result via method I could also be obtained, in practice, by automatic computation, but the evaluation steps of method II are, in our opinion, simpler and therefore easier to program. With method I, one applies equation (A.44) with, for example, A = S(i) + S(2) and B = S(3 ) to obtain (S(l) + 5(2) + S(3)) © S(21) = [(S(l) + 5(2)) © «(21)] [«(3) © 5(0)] + [(«(1) + 5(2)) © 5(0)] [«(3) © s(21)] + [(S(l) + S(2)) © 5(2)] [S(3) © 5(1)] + [(3(i) + 3(2)) © 5(i)] [s{3) © S(2)] + [(S(i) + 5(2)) © 5{12)] [3(3) © S(i)] + [(5(1) + S(2)) © 5(i)] [S(3) © S(i2)] (A.60) Applying equation (A.44) to the first factor of each term in the preceding equation, one is left with plethysms of “simple” S-functions which can be evaluated using one’s favourite algorithm. The result is sums and products of sums of S-functions. Repeated applications of the Littlewood-Richardson rule then generates the result (s(i) + S(2) + 5(3)) © S(21) = 5(81) + S(72) + 5(63) + 5(621) + 5(54) + 5(531) + S(432) + 5(8) + 2S(7i) + 3S(62) + 5(612) + 3S(53) + 2S(52 i) + S(4?) + 2S(43i) +5(4 2 2) + 5(322) + 2S(7) + 45(61) + 5S(52) + 2S(5i2) + 4S(43) + 3S(42i) + 2S(32i) + S(322) + 25(6) + 5S(5i) + 5S(42) + 2S(4i2) + 25(32) + 3S(32i) + 25(5) + 4S(4i) + 3S(32) + 2S(3i2) + S(22i) + 5(4) + 2S(3i) + S(22) + S(2J2) + 5(21) (A.61) Note that the above result is valid for U(n) in general, i.e. for n > 3, since no S-functions in more than 3 parts appear in the expansion. With Method II we proceed as follows. First S(i) + S(2) + S(3) is expressed in terms of m-functions and S (2 i ) in terms of p-functions, (5(1) + 5(2) + 5(3)) @5(21) = (™ (1) + "1(2) + "l(l2) + m (3) + m (21) + ^ (l3)) (P(l3) “ P(3)) (A.62) Use of equation (A.47) yields (5(1) + S(2) + 5(3)) © S(2l) = | (m (l) + m (2) + m {l2) + m (3) + I"(21) + ?"(13) — | ("2(3) + "1(6) + m (S2) + m (9) + "1(63) + m (S3) The next step is to evaluate the first term on the r.h.s. of the above equation by applying equation (A.31). At this point the desired number of indeterminates has to be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 138 specified. Since we are interested in a 17(3) result, we restrict the m-function labels to at most three parts. Finally we collect like terms in the resulting m-function expansion and by means of equation (A.20) convert them to S-functions6. It is clear from this example that the algorithm presented here does not rely on recurrence relations but rather follows very simple rules extremely convenient for auto matic computation. The rules of this algorithm are exactly the same whether one has to determine the plethysm of simple or compound S-functions. The Maple procedure implementing method II of plethysm is > Splethysm:=proc(dim::nonnegint,S1::list(list),S2::list) local SS2,SS1, SR; SSI:=M_content(dim,Sl); SS2:=P_content(S2); SR:=pleth(dim,SSI,SS2);RETURN(Schur_list(dim,SR)); end: Thus, the command to evaluate the plethysm (s(i) + S(2) + «( 3 )) © S(2 i) , and the corre sponding output are > Splethysm(3,[[1] ,[2], [3]] , [2,1]); [ [8, 1, 0, 1], [7, 2, 0, 1], [6, 3, 0, 1], [6, 2, 1, 1], [5, 4, 0, 1], [5, 3, 1. 1], [4, 3, 2, 1], [8, 0, 0, 1], [7, 1, 0, 2], [6, 2, 0, 3], [6, 1, 1, 1], [5, 3, 0, 3], [5, 2, 1, 2], [4, 4, 0, 1], [4, 3, 1, 2], [4, 2, 2, 1], [3, 3, 2, 1], [7, 0, 0, 21, [6, 1. 0, 4], [5, 2, 0, 5], [5, 1, 1, 2], [4, 3, 0, 4], [4, 2, 1, 3], [3, 3, 1, 2], [3, 2, 2, 1] [6, 0, 0, 2], [5, 1, 0, 5], [4, 2, 0, 5], [4, 1, 1, 2], [3, 3, 0, 2], [3, 2, 1, 3] [5, 0, 0, 21, [4, 1, 0, 4], [3, 2, 0, 3], [3, 1, 1, 2], [2, 2, 1, 1], [4, 0, 0, 1], [3, 1, 0, 2], [2, 2, 0, 1], [2, 1, 1, 1], [2, 1, 0, 1] ] Clearly, the entries in the call-command Splethysm(dim,Sl ,S2) are dim=3, where dim is the chosen number of indeterminates; SI = [[1] , [2] , [3]], a list of lists; each list-element corresponds to an S-function on the l.h.s. of the plethysm operation; S2 = [2,1], a simple list which stands for the S-function on the r.h.s. of the plethysm operation. 6Note here that, as mentioned before in this paper, having to construct the Kostka matrix in order to retrieve the required inverse Kostka numbers is not as economical as using equation (A.20). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 139 Note that the entry-lists do not require the zero parts of the S-functions to be specified but, in the output, the S-functions are labelled by lists with dim+1 number of parts. The last element of each list gives the m ultiplicity of the corresponding S-function. The procedure Splethysm relies on other Maple procedures, namely M_content which expresses the S-functions in terms of m-functions, P_content which expresses an S- function in terms of p-functions, pleth which evaluates the plethysm of m-functions with p-functions, evaluates the product of series of m-functions (whenever necessary) and col lects like m-functions together, and finally Schur_list which converts the m-functions back to S-functions. As w ith any algorithm for plethysm, run tim e increases rapidily w ith the dimensions of the S-functions involved. W ith a 100 MHz Pentium the plethysm above mentioned takes 12 seconds. An important point of this method is that one has the ability to establish, a priori, the affiliation of the character sM, (in © s„), i.e. whether it belongs to U(2), or U(3), etc., so that one can specify the number of indeterminates and considerably simplify the calculations and reduce running time. For example, with the same 100 MHz Pentium, the plethysm S( 22) ® S(8) takes 11 seconds for r = 2 (U (2)) indeterminates, 48.5 seconds for r = 3 (17(3)), and about 20 minutes for r = 4 (17(4)). Note though that the simplification introduced by establishing from the beginning the maximum number of parts of the resulting S-functions does not result in any loss of accuracy. In conclusion we have succeeded in giving a method that treats the plethysm of com pound S-functions (linear combinations or products of S-functions) on the same footing as the plethysm of simple S-functions. There is no need, in this method, to resort to the use of intricate equations in order to take care of the fact that S-function plethysm is not distributive on the left with respect to addition, subtraction and multiplication. The key point of the method is the fact that the plethysm of a monomial symmetric function by a power-sum symmetric function is still a monomial symmetric function, which does not hold true in general for S-functions. Clearly, the product of two m-functions is a series of m-functions; thus by reducing the S-functions to m-functions the plethysm reduces to a simple multiplication of m-functions, which can be performed without regard for their S-function origin. The S-function content of the plethysm is recovered at the end by converting the final m-functions into S-functions, using again a simple algorithm. The method requires only algorithms for expanding S-functions in terms of m-functions and vice versa, and for evaluating products of series of m-functions. These algorithms are given in Sections III and IV respectively. Use of this method is of great advantage for evaluating plethysms of characters of groups other than the unitary groups or plethysms of finite series of S-functions. Maple procedures to carry out the calculations entailed by these algorithms have been constructed and the whole package will appear shortly in the literature. The procedure to evaluate the outer product of S-functions is also part of the package. Details w ill be left to the forthcoming publication. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 140 Acknowledgements The authors are very grateful to Professor D J Rowe for fruitful discussions and his critical reading of the manuscript. We would also like to thank Professor B G Wybourne for correspondence on the matter of footnote 6. SD would also like to thank Dr C Bahri for helpful discussions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. Appendix A — The Kostka numbers and representation theory The physical interpretation of Kostka numbers can be made clear in terms of repre sentation theory by means of the following claim. Claim: If A is a highest weight for some U(n) irrep then the S-function (character) for this irrep is given by + ^0*0 = ^ K Xvm v(x) , (A.63) where the sum J2+ is restricted to dominant integral weights and K Xu is the m ultiplicity of basis states of a weight v in the irrep. Note that a dominant integral weight is one having the property that > v2 > • • • > vN > 0. In the language of Lie algebra structure theory, such a weight belongs to the positive Weyl chamber of weight space. Proof: An S-function sx(x) for a unitary group U (n) evaluated at x = (x x,..., xn) is the trace over a basis for the irrep of highest weight A of the matrix T x (a;) representing the diagonal U (n ) matrix with diagonal entries ( x i,..., x n). Under the transformation T x(x), a state \v) of weight u = (vX) v2,...) and multiplicity indexed by a transforms as T (x ) : |a, v) —» \a, v) x xvyx v2 .... (A.64) It follows that sx = Y ^ ,x Tx22---, (A.65) where the sum is over all weights of the irrep. Now observe that, if v is a weight of a U (n ) irrep, the weight obtained by permuting all the parts of v = (i/i, v2,...), viewed as a partition of |A|, is also a weight for the irrep. The set of all weights obtained by permuting the parts of a given weight lie on a Weyl orbit. Such an orbit is characterized by any one weight in the orbit. Moreover, every orbit contains just one weight in the positive Weyl chamber. Next observe that the contribution to the S-function coming from all the weights on a single Weyl orbit is the sum of the distinct terms obtained by permutations V of the subscripts: = ^ / 'P x T x 22 ■ ■ ■ • (A.66 ) v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 141 It follows that the S-function for an irrep is the sum of such m-functions weighted by the multiplicities K \ v. Appendix B — Multiplication of m-functions The product of two simple m-functions is defined as mamp = Y J , 7a/3 m i > (A.67) where { 7 } stands for the set of distinct (and ordered) partitions obtained by adding partition a to the partitions derived from {3 by permuting its parts in all possible, but distinct, ways: dim(ma.) (A.68 ) Aw) n7 dim (m 7) In the case the product of monomial functions involves multiplicative coeficients, we define cam Q cpmp = T 7a5 m 7 , (A.69) { 7 } where nr . . 7 . _ dim (ma) lK ^ — °aCP Aa/3 — C“C/3 n 7 'j ’ (A.70) The product of a sum of two monomial functions by another monomial function, is: (ca im ai + ca 2Tna2) cprrip = ^ ^ ^71 “h ^ ^ mi 2 { 71} { 72} = ^7afl m7 5 (A.71) { 7 } where the set of partitions { 7 } is the union of the sets { 71 } and { 72 }- T7Q1^ + T7*2/? if 7 € {7 1 } 11(72} T7a^ if 7 € (7l) (A.72) T 7a20 if 7 € {72} The generalization to a sum of N terms is straightforward 7a/? m7 (A.73) v i J { 7 } with £ i T7a,./? if 7 €(1(7*} T 7a/? = (A.74) T 7c^ if 7 € {7 i} Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 142 Now the product of two series of monomial functions is ^ 2 iCah maii j ( Coti2m “ i2 ) — XZ T 7h, ^ *1 / \ *2 / {7i1,i2} ^ ^ T 712 W'7 i 2 > (A.75) { 712} where T 712H lf 7 G ™ 7 il’^ . (A.76) T 7n,i2 if 7€{7n,i2} Or equivalently, T712 = ^7j1,i2^7l2{7ii,i2} 5 (A.77) * 1,*2 where 57{7il i<2} = 1 if the partition 712 appears in the set { 7 *ljt2} and zero otherwise. The product of three series of monomial functions can be obtained by making use of the previous result I Ylchm<*n ) [J2Ci2m«i2 ) ) = 5Z T 712m7(12) \ *1 / \ *2 / \ *3 ) V { 7 1 2 } / *3 ^ 7l2m 7(12) I Ci3rnau — ^2 m 7 l 2 , j 3 < { 712} / *3 {712,i3} = ^ ^ ^7123 mil23 (A.78) { 7 1 2 3 } w ith ^7123 = (A.79) *12*3 dim (m 7l„. ) ^712,1-3 = T 712C*3«712,*3 d J m ^ J ( A ' 8 0 ) _ dim(maii) 7 i i , < 2 dim (m 7l2) ' ( • ) Finally the product of p series of monomial functions can be w ritten as ( y ' c*im a i, j f y yc*2m Q>o j ■■■ 1 J = y , m n2...p > (a .8 2 ) \ *1 / \ *2 / \ ip J { 712. ..p } Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 143 where the coefficients T 71,. , = ^7l2...p-l,»p^712...p{7l2...p-l,ip} (A.83) *12...p-l,ip can be found recursively. References • [1] Littlewood D E 1950 The Theory of Group Characters and Matrix Representa tions of Groups , second edition (Oxford: Clarendon Press) • [2] Littlewood D E 1936 Polynomial concomitants and invariant matrices, J. London Math. Soc. 11 49-55 • [3] Wybourne B G 1970 Symmetry Principles and Atomic Spectroscopy (New York: Wiley-Interscience) • [4] Carvalho M J 1990 Symmetrised Kronecker products of the fundamental repre sentation of Sp (n ,R ), J. Phys. A : Math. Gen. 23 1909-1927 • [5] Wybourne B G 1992 The representation space of the nuclear symplectic Sp(6 , R) shell model J. Phys. A : Math. Gen. 25 4389-4398 • [6 ] Ng C N M and Carvalho M J 1996 The Sp(3, R) representations of an A-particle nucleus, Comp. Phys. Comm. 96 288-300 • [7] Grudzinski K and Wybourne B G 1996 Plethysm for the non-compact group Sp(2n, R) and new S-function identities J. Phys. A : Math. Gen. 29 6631-6641 • [8] King R C and Wybourne B G 1998 Products and symmetrized powers of irre ducible representations of Sp(2 n ,R ) and their associates, J. Phys. A : Math. Gen. 31 6669-6689 • [9] Butler P H and King R C 1973 Branching rules for U(iV) D U(M) and the evaluation of outer plethysms, J. Math. Phys. 14 741-745 • [10] Chen Y M, Garsia A M and Remmel J 1984 Algorithms for plethysms, in Combinatorics and Algebra , edited by Curtis Greene, Contemporary Mathematics, Volume 34, (Providence: American Mathematical Society) 59-84 • [11] Yang M 1998 An algorithm for computing plethysm coefficients, Discrete Math. 180 391-402 • [12] MacDonald I G 1995 Symmetric Functions and Hall Polynomials , second edi tion (Oxford: Clarendon Press) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix A . 144 • [13] Proctor R A 1994 Young tableaux, Gelfand patterns, and branching rules for classical groups, J. Algebra 164 299-360 • [14] Carvalho M J 1993 S-function series revisited, J. Phys. A : Math. Gen. 26 2179-2197 • [15] M uir T 1923 The Theory of Determinants Volume 4 (Reprinted 1960, New York: Dover Publications Inc.) • [16] King R C 1975 Branching rules for classical Lie groups using tensor and spinor methods, J. Phys. A 8 429-449 • [17] Littlewood D E and Richardson A R 1934 Group characters and algebra, Roy. Soc. Lon. Phil. Trans. A233 99-141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B A MAPLE program for calculations with Schur functions M.J. Carvalho, S. D’Agostino Computer Physics Communications 141 (2001) 282-295 Received 26 March 2001; accepted 28 June 2001 A b s tra c t We describe an interactive program, to be used in the M APLE environment, for calculations involving Schur functions. The program performs Schur func tion operations of interest to applications in many-body physics: outer prod uct, internal product, division and outer plethysm. Essential to the pro gram is a new algorithm for the multiplication of monomial symmetric func tions, which is used to implement the evaluation of outer product and outer plethysm. The same algorithm is extended to apply to the division of mono mial symmetric functions, with a novel use of partitions containing negative integers, useful for implementing-the evaluation of division. In particular, the evaluation of outer plethysms involving sums, differences, and products of Schur functions with this program is as simple as those involving only single Schur functions. The program can also be used to expand a Schur function in terms of monomial symmetric functions or expand a monomial symmetric function in terms of Schur functions, i.e. to calculate the Kostka and inverse Kostka numbers. The calculation of Kostka numbers makes substantial use of Gel’fand patterns. The number of variables in which Schur functions or monomial symmetrical functions are expressed can be specified in advance by the user, which results in significant reduction of calculation time. PACS codes: 0220 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 146 PROGRAM SUMMARY Title of program: SCHUROPERA Catalogue identifier. AD PF Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer. Pentium 933 MHz, Compaq Presario 1260, Pentium 300 MHz Operating system: Windows 95/98 High speed storage required: none Peripherals used: none Number of lines in the program: 353 (including examples) No. of bytes in distributed program, including test data, etc.: 15 735 Distribution format: tar gzip file CPC Program Library subprograms used: none Keywords: Schur functions, outer product, internal product, plethysm, unitary group, symmetric group, character table Nature of the physical problem: most common operations involving Schur functions, which are widely applied, in particular to many-body physics Method of solution: The outer product, skew division, and outer plethysm of Schur functions rely on an algorithm for the multiplication and division of monomial symmetric functions [1]. For the internal product of Schur functions, extensive use is made of the MAPLE library program “combinat” [2]. Restrictions on the complexity of the problem: main limitation is running time Typical running time: Running time depends both on the operation performed and on the computer processor type. Elapsed times for specific operations are reported in the test runs. References [1] Carvalho M J and D ’Agostino S 2001 Plethysms of Schur functions and the Shell model, J. Phys. A : Math. Gen. 34 1375-1392 [2] M APLE 6 , ©2000, Waterloo Maple Inc. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 147 LONG WRITE-UP B.l Introduction This paper documents the program SCHUROPERA, which enables a user to carry out all of the standard operations involving Schur functions in a MAPLE environment. Schur functions (S-functions for short) are characters of the unitary irreps of unitary groups (and the characters of finite-dimensional irreps of general linear groups) [ 1]. The groups of interest in many-body physics are, besides the symmetric groups, the classical compact Lie groups (general linear, special linear, unitary, special unitary, or thogonal, special orthogonal, and unitary symplectic), as well as the non-compact sym- plectic Lie groups. The characters of the unitary irreps of all of the above mentioned continuous groups can be expressed in terms of S-functions, and vice versa. Calculations of physical relevance involving characters of such irreps can therefore be transferred to the realm of S-functions, and the results expressed back in terms of irreps of the group in question. Thus the operations involving S-functions are basic to calculations involving characters of unitary irreps of the aforementioned continuous groups and have funda mental physical importance for descriptions of many-particle states. This motivates the need for a program that efficiently effects the calculus of S-functions. Such programs exist (SCHUR, for example [ 2]) but we believe that this is the first one that is intended to be used in a MAPLE environment. The program, which incorporates newly developed algorithms (using monomial symmetric functions) for outer product, skew division and outer plethysm [3], is extremely straightforward to use. The symmetric functions of interest in this paper are S-functions, monomial symmet ric functions (m-functions for short) and power-sum symmetric functions (p-functions for short). Their definitions are as follows. Consider a fixed number r of variables ( indeterminates) xi, #2, • • •, %T and a partition A = (Ai, A2,..., Afc). The S-function labeled by A and denoted by s\ (or alternatively by {A}) is expressed in terms of the indeterminates by I x M+r-t (B.l) where s and t index rows and columns respectively of the r x r determinants. The monomial symmetric function labeled by A and denoted by m \ is defined as mx = Y ,xilx 2 2-'- 4 k (k where the sum includes all distinct terms obtained by all possible permutations of the subscripts i of the indeterminates x*. The number of terms in the sum (which we call Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 148 the dimension dim(raA) of the m-function), is dim(raA) = j- , (B.3) l l j Pi- where p* = Vi for * > 1, (ui is the multiplicity of the part A* in A) and p 0 = r — k. Finally, the power-sum symmetric function labeled by A and denoted by pA is the product p \ = P\iP \2 • •' Pxk where Pj, w ith j = A,, is the sum of the j-th powers of the r indeterminates: r Pj = J 2 Xi=m(i)- (B ‘4) i= 1 Further information about partitions and symmetric functions can be found in [4]. B.2 Transitions Among Symmetric Functions Operations with S-functions are sometimes facilitated by expressing them in terms of other symmetric functions. For our purpose in this paper we shall only need to discuss the expansion of S-functions in terms of m-functions and p-functions, and the expansion of m-functions in terms of S-functions. More complete discussions of relations among symmetric functions can be found in [4], For A h n, the S-function sA is expressed in terms of p-functions as follows 1 where p = (pi, p2,...) labels the conjugacy classes of Sn, pp = n , Pm are p-functions, and Xp are the components of the character of the irrep A of Sn. The quantities zp are given [4] by zp = ul \ r i y2\2^---un\rUr , (B.6 ) where ^ counts the number of parts of A that are equal to i. On the other hand, the expansion of the S-function sA in terms of m-functions is s\ = 'Y! K** mi* ’ (B -7) M=|A| where p is a standard partition of |A|. The coefficients which are either positive integers or zero, are known as the Kostka numbers [4]. When the partitions A and p are 1The remarkable relationship between the unitary and symmetric groups, known as the Schur-Weyl duality, leads to a direct relationship between a character xX of the symmetric group and the corre sponding character s\ for a unitary group. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 149 placed in reverse lexicographical order, then the arrays formed by the Kostka numbers are upper-triangular. Inversely, an m-function, raA, is expressed in terms of S-functions according to m A = \ K ^ s a . (B.8) crh|A| where the coefficients K x^ (the inverse Kostka numbers), can be either positive or neg ative integers. The number of terms in the sum (B. 8) depends on the number of in- determinates, but is certainly less than or equal to the dimension of raA (cf. equation (B.3)). B.2.1 Calculation of the Kostka numbers Several methods are given in the literature for the determination of the Kostka numbers K \fi. Typically they are calculated by combinatorial means involving Young diagrams. However, for automatic computation, it is more efficient to identify and enumerate the partitions // in (B.7) by means of Gel’fand patterns. For a given 5-function (in r indeterminates), labeled by A = (Ai, A2,..., Afc) (w ith r > k), the m-functions appearing on the r.h.s. of (B.7) can be identified by means of triangular arrays of integers (the Gel’fand patterns) [5] Sll Sl2 Sl3 ' ■ ' Sir—1 Sir S21 S22 S23 ’ ‘ ' S2r-1 S31 S32 ' ‘ ' S 3r-2 (B-9) Srr whose entries are subject to the following conditions: Ai for i < k 9 li = (B.10) 0 for k < i < r and 9k—l,i — 9ki ^ 9k—l,i+ l ■ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 150 The parts of each partition \i (except the last one) are given by the differences of the sums of the entries in two successive rows of the array, i.e. r—i+1 r—t = y ^ 9u ~ y ] 9i+i,i 1=1 1=1 flr = 9rr ■ To ensure that only standard partitions (i.e. the leading term of each m-function m are obtained, one further imposes the condition r —i+1 r —i r —i r —i—1 9il - > 'Y^9i+l,l ~ 5Z 9i+2,l • (B.13) 1 = 1 1 = 1 1=1 1=1 As an example consider the S-function S( 2i) in three indeterminates. The triangular patterns that satisfy the above given conditions are 2 1 0 2 1 0 2 1 0 1 0 2 0 1 1 (B.14) 0 1 1 and the partitions n are respectively (21), (111), (111)- Thus the result: S( 2i) = m(2i) + 2 m ( m ) . B.2.2 Calculation of the inverse Kostka numbers A practical, easily programmable procedure for determining the inverse Kostka numbers requires only two straightforward steps conveyed by the following formula 2 = J 2 sa = J 2 K I * S° ■ (B -15) A ° where the first sum runs over all distinct partitions A generated from A by permuting its parts in all possible ways. Clearly, only one of the partitions A is standard. Application of the S-function modification rules [9] to the non-standard S-functions leads to sa and the sought-after inverse Kostka numbers. 2Note that this procedure arises from multiplying an m-function by a Vandermonde determinant and rearranging the result as discussed in [1, 6], This procedure can be traced back at least as far as Muir in 1882; see [7, pp 150-151]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 151 As an example, consider the m-function m( 53i) in three indeterminates. First, we have that ^ ( 5 3 1 ) = 5(531) + 5(513) + 5(351) + 5(315) + 5(153) + 5(135) i (B.16) and since, by the m odification rules, 5 (3 1 5 ) = S ( i 5 3 ) = 0 5(513) = —S(522) 5(351) = —5(441) 5 ( 1 3 5 ) = 5 ( 3 3 3 ) i (B.17) then the resulting S-function expansion (in three indeterminates) is wi(531) = 5 ( 5 3 1 ) — S(522) _ 5(441) — 5(333) • (B.18) B.3 The Multiplication of Symmetric Functions Each operation involving S-functions corresponds to an operation on unitary represen tations of G L(n) or its subgroups. [ 8] In particular, the outer product of S-functions corresponds to the tensor product of unitary irreps of U(n) or G L(n), and the internal product (also known as the inner product) corresponds to the tensor product of Sn irreps. The physical importance of the outer product and the internal product of S-functions in the description of many-particle states is illustrated next. First, suppose that a state of Np protons is specified by sx and a state of Nn neutrons is specified by sp, so that A F Np and /i I~ N n. Assuming that U (n ) is the underlying group of the model both s\ and Sp are representations of U (n). A state of the combined system of Np + Nn particles is specified by an 5-function sa that occurs in the expansion of the outer product 5^5^ = ^ ^ r \ p(rSc , (B.19) <7 where each partition a in the expansion is a partition of Np + N n and s till an irrep of the same unitary group U (n). The relevance of the internal product of S-functions in physical applications stems from the fact that we can also interpret this product in terms of irreps of unitary groups. For example, suppose that the orbital angular momentum wave function of a system of N particles is specified by the irrep s\ of U (2 j + 1), and the spin angular momentum wave function of the same system is specified by the irrep sp of U(2s + 1). Then a total wave function of the system is specified by an S-function sa in the expansion of the internal product = kxn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 152 where each irrep s„ is an irrep of U ((2 j + l)(2s + 1)). Note that each partition A, p and a is a partition of N , since each irrep labels the same system of N particles. Noting also that the Kronecker product of Sn irreps satisfies as well the condition that all irreps are labeled by partitions of N , one may guess that the internal product can serve for both the Kronecker product of Sn irreps and the shell model situation described in this paragraph. The fact that the internal product actually does apply to both is a manifestation of Schur-Weyl duality. The symmetrized (outer) product of p copies of the same S-function, also called (outer) plethysm, is another S-function operation that is of importance and commonly used in physical applications. Consider single-particle states labeled by s\ corresponding to an irrep of U (n); then states of N identical particles with permutation symmetry [v], v \- N , are labeled by an S-function sn occurring in the expansion of the outer plethysm © s„ = (B.21) a Note that in equation (B.21), v labels an irrep of SN and the sa label irreps of U(n). B.3.1 Multiplication and division of m-functions Symmetric monomial functions are specified by partitions; thus, to effect the product or division of two m-functions one has to manipulate the corresponding labeling partitions. Let’s define the result of the addition and subtraction of two partitions (pi,p2, • • •) and (iq, h>2, ■■■) as being the partition whose parts are (p\ ± rq, /r 2 ± z'2, • • •)• fo r these op erations to be meaningful, it is necessary that both partitions have an equal number of parts; if they don’t, then one increases the number of parts of the shortest one by adding enough zeros at the end. However one should note here that, for division, it is further required that the partition to be subtracted, n, have no more non-zero parts than the number of non-zero parts of the partition p. The multiplication and division of two m-functions are then defined as m amp — ^ 2 1 7 m~, , (B.22) 7 and E Iry! my , (B.23) i where the partitions 7 , 7 ' result from adding to or subtracting, respectively, from a all distinct partitions obtained by permuting in all possible ways the parts of /3. Clearly, all m-functions involved in (B.22) and (B.23) are functions of the same r indeterminates, i.e. have the same number of total parts. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 153 The coefficient with v = 7 , 7 ' is given by r dim (m Q) /R94x — n u . , \n .Z 4 ) dim (m „) where n„ is the number of times the same partition v appears in the process of adding or subtracting partitions referred to above. Note that as a result of the division of m-functions one may obtain m-functions labeled by partitions with negative numbers. Since the objective of defining this operation is to serve as an intermediate step leading to the division of S-functions one should not hastily discard m-functions with such unorthodox labels. Indeed, one doesn’t associate an actual polynomial w ith an m -function that is labeled by a partition that has negative parts. This is just an artifice to effect the division of S-functions by using m-function operations, by analogy w ith the m ultiplication process. For example the product m pnjm pn) in 17(4) (r = 4 indeterminates) gives: 777(211)777(in) = 771(3220) + 2 777(3211) + 3 7 7 7 (2221) 5 (B .2 5 ) and the result of the division 777 (211)/777(m), also in 17(4), is 777(211)/777(111) = 3 777(1000) + 2 T O (n o -l) + 777(200-1) j (B .2 6 ) B.3.2 Outer product and division of S-functions The standard method for evaluating the outer product or skew division of S-functions is the well-known Littlewood-Richardson rule [11], which is a set of directives to be applied to the corresponding Young diagrams. Although application of these rules is feasible in simple cases, their execution for S-functions that are labeled by partitions of large numbers is rather complex. The resolution of the outer product/division of two S- functions via the product/division of m-functions [3] is the one we exploit in the MAPLE procedures. Given two S-functions s^, sv, their outer product can be easily determined by expand ing each of them in terms of m-functions, evaluating the multiplication of the m-functions and converting the resulting m-function expansion back to S-functions. That is, Sfj,Sv (B .2 7 ) ^ 2 T 7 7777 (B.28) 7 (B .2 9 ) <7 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 154 where A = E A , A b « ) . (B-30) y w ith T7.. = K m i Kypj L hj and $7{7j.} = 1 if the partition 7 appears in the set { 7 ^ } and zero otherwise. Similarly, the division of two S-functions s^, s„, (with \u\ < |/x| and p, < >u*), is given by s j s — ( V K m i m a^ j j A ’ly . m p. ^ = ^ ] ^nai Kvpj f ^ai /'mPj Oti , p j — ^ *] Y y my y — E E ATy^Yy V , (B.31) where Y y is defined as in (B .3 0 ). B.3.3 Internal product of S-functions The internal product sa o sT of the S-functions sa and sT was devised by Littlewood [9, §3 .11] to correspond to the decomposition of the tensor product of simple characters of a symmetric group. That is, the internal product of S-functions sa o sT = ^ (B .3 2 ) corresponds to the decomposition of symmetric group characters (B .3 3 ) A* In the previous two equations, <7, r, and n are partitions of the same number N, and the characters belong to Sn- Also, the coefficients kaTn are to ta lly symmetric, so no independent operation of internal division (analogous to the division that is associated with outer multiplication) exists, as it would be just the same as internal multiplication [8, §3]. Besides the uses already mentioned, the internal product of S-functions is also useful in determining certain branching rules of classical Lie groups [ 8]. Some potentially useful properties of the internal product of S-functions are [9]: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 155 • if' Sp o s r — ^ '.y fcaTtiSfi then S& o S f — ^ X and s $ o S f — s ff o s T • Sjy O S(j — Sfj O Sjv ~ Sff • Sin O S j = Sf f O S^iV = S • sa o sa contains sjv • s$o sa contains Sj w • internal product is distributive over addition on the left and right. • (S\Sv,) OSu = '£ a (SX O Sv/a) (S„ O S where the symbol a signifies the conjugate of the partition a, and each S-function label is a partition of N. The standard method of evaluating the internal product of S-functions, which avoids the construction of character tables, is a recursive method due to Littlewood [9, §3.11], [10, p 40], which exploits the properties given above. On the other hand, the program SCHUROPERA determines the internal product of Schur functions by determining the corresponding product of symmetric group characters, taking advantage of the fact that these are readily available in the M APLE library “ com binaf. One first represents the rows of the appropriate character tables as vectors. Then the Kronecker product of two characters is obtained by forming a vector whose components are the products of corresponding components of the character factors. For example, consider the character table of S3. One defines vectors to represent the rows of the table: = (1,1,1), x ^ = (2,0,-1), and x ^ = (1, —1,1)- Consider then the Kronecker product [21] x [21]; its character is x ^ * — (2,0,—1) • (2,0,-1) = (4,0,1). Now one wishes to express the result in terms of the simple characters or, in other words, to determine the coefficients ka in the expansion (4,0,1) = /c(3)(1,1,1) + k(2i)(2, 0, - 1 ) + k{13)(1, -1 ,1 ) . (B.34) It is easy to verify that in this case, fy3) = 1, 1) = 1 and k(i 3) = 1. The corresponding S-function decomposition is then 3(21) 0 3(21) = 3(3) + S(2J) + S(!3) (B.35) Once again, the problem of solving a vector equation such as equation B.34 is easily handled by the M A P LE library package “ linalg ”. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 156 B.3.4 Outer plethysm of S-functions The outer product of k copies of an S-function s\ can be decomposed into a sum of sets of terms, 'SxSxl 6'A-= ^ 2 , (B.36) k copies ^ where the sum on the right side extends over all partitions of k, and is equal to the dimension of the S* irrep labeled by ji. The decomposition of each set in terms of S-functions defines the operation of plethysm, i.e. S\ © ^ ^ A\[,asa (B.37) ahnk which gives a sum of S-functions, all of weight nk, with non-negative integer coefficients A A fia - This operation is of great relevance in many-body physics. I f the states of each indi vidual particle are described by s\ then the states of a /c-particle system of permutation symmetry (/x) are described by s\ © sy Note that a given S-function may appear in different plethysms, so S-functions do not characterize the symmetry classes of tensor powers. In contrast with outer product, plethysm of S-functions is not commutative or distributive on the left over outer product, addition and subtraction. Some of the algorithms for plethysm reduce the operation s\ © to a plethysm of S-functions with p-functions [ 12], the latter arising from the expansion of in terms of p- functions. The algorithm on which the MAPLE procedure for plethysm, here presented, is based, involves one further expansion. By first expressing the S-functions on the left side of the plethysm sign @ in terms of m-functions and the S-function on the right side in terms of p-functions, the plethysm of an S-function is then reduced to the plethysm of a series of m-functions w ith p-functions. Such plethysms are straightforward, since the plethysm of an m-function with a p-function is still an m-function, m n © P j = i (B .3 8 ) where j./j, means that each part of /x is m ultiplied by j (that is, if /x = (/xj,..., /x*,) then j. f i = (j/xi,..., i/Xfc)). The resulting series of m-functions is then expressed back in terms of S-functions. This method [3] makes it possible to evaluate the plethysm of a sum or product of S-functions (compound S-function) by another S-function in exactly the same manner as the evaluation of the plethysm of a single S-function by another. Note that the traditional methods have to make use of cumbersome recursion relations to evaluate plethysms involving sums or products of S-functions [9]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 157 Thus, for the plethysm of single S-functions, we have that ®A © 8V — ^ ^ Axua = Ylxpzpls*©I[pf>* P i = e x fe p 1 n ® ppi • (b -39) P * where the S-function sv was expanded in terms of the p-functions pp — J |.pPi according to sv = YlpX^zplPpi and use was made of the fact that plethysm of S-functions is distributive on the right. Expanding now the S-function s\ in terms of m-functions sa = K W n*j (B -4°) i and making use of m p © p j = mj.p, Kp ) ( \ S\@SU = J2XpZp 1 n ( E K ^ j m Pi.aj j , (B.41) P i \ j / one is faced with the very simple, though repetitive, task of evaluating the product of m-functions and keeping track of the number of times they appear [3]. The final step is to convert the series of m-functions back to S-functions using the method outlined in section B. 2 .2 . One then has a 7 and Axua = J 2 r , K - a1 . (B.43) 7 The plethysm of, for example, a sum or difference of S-functions is mechanically not more complicated than the procedure for the plethysm of a single S-function, as illustrated in the following equation. (sA ± sp) © sv = E ^ n ( E K \ a m pi-a i • (B.44) p i a |8 Note that generalizing equations (B.41) or (B.44) to a plethysm with a sum of S-functions in place of sv is trivia l, since plethysm is distributive on the right. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 158 B.4 Program The SCHUROPERA package consists of several M APLE procedures organized in 7 units tailored to execute particular operations with symmetric functions. Since S-functions, m-functions and p-functions are all labeled by partitions, any one of these functions is identified, in the MAPLE procedures, by a list of integers. In some cases the labeling list contains one extra element at the end (not necessarily an integer), which specifies the coefficient associated with that symmetric function. For example the S-functions resulting from the outer product of two S-functions are presented in that format with the last element of each list giving the m ultiplicity of the corresponding S-function in the expansion. The number of indeterminates in which each symmetric function is expressed, that is the number parts of the labelling partition, is denoted by dim and is an input parameter in almost all procedures. For the sake of sim plicity these Maple units have not been incorporated into a M APLE table so that the reader has the freedom to make use of them separately or reassemble them in any other convenient way. The M APLE libraries that this package requires are only linalg and combinat. Unit 1 is headed by the procedure m_content, which gives the expansion of a particular S-function in terms of m-functions. The call command is m_content(dim,L) where L is a list standing for the labeling partition of the S-function and dim is the desired maximum number of parts of the resulting m-functions in the expansion (in shell model applications dim is the dimension of the unitary group of interest). The output is a list of lists each of which specifies an m-function. The output lists have dim +1 elements, with the last one giving the coefficient of the particular m-function in the expansion. For example: > m_content( 6 , [4,3]); [[2, 1, 1, 1, 1, 1, 9], [2, 2, 1, 1, 1, 0, 6 ], [2, 2, 2, 1, 0, 0, 4], [3, 1, 1, 1, 1, 0, 4], [3, 2 , 1, 1, 0, 0, 3], [3, 2 , 2 , 0, 0, 0, 2 ] , [3, 3, 1, 0, 0, 0, 2], [4, 1, 1, 1, 0, 0, 1], [4, 2, 1, 0, 0, 0, 1], [4, 3, 0, 0, 0, 0, 1]] In other words, in U (6 ) (no more than 6 parts) S(43) = m (43)+m (42i)+m (413)-t-2m (321)-|-2m(322)-|-3m(32i2)-l-4m(314)+4m (23i)+6m (22i3)-t-9m (2i5) (B.45) The procedure m_content relies on the following procedures: • a d ju s t(d im ,L ) whose role is to add enough zeros to the end of the input list L so that the total number of parts of the list it returns is equal to dim. Clearly this is necesary so that one only has to specify the non-zero parts of the entry list L. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 159 • classes (dim, N,n) which constructs the partitions of the number N (where N is the sum of the elements of the list L) satisfying the condition that each partition’s largest element is not larger than n, the first (largest) element of the input list L). Use is made of the MAPLE library routine that constructs such partitions but a further rearrangement is necessary, which is done by ord_adj. • ord.adj (dim.L) which for each input list L, with parts in ascending order, returns a list with parts in descending order and with zeros added at the end, so that the length of the output list equals dim. • rows (dim,L.C) constructs possible candidates for the the elements of the second, third, etc. rows of the Gel’fand patterns (see section B.2.1) where C is an auxiliary list that calls on the next procedure. • element(s,c), Check(II) check(Ll,L2) checks if the row elements constructed in the previous procedure are possible or not; i.e. these procedures ascertain that the conditions given in section B.2.1 are satisfied. Still in unit 1 is the procedure M_content (dim , LL), a generalization of m_content (dim, L) that accepts a list of S-functions as input, instead of a single one. Thus it requires, be sides m.content, the procedure Add_list, which gathers in a single list of m-functions the various lists of m-functions arising from the expansion of each S-function. The pro cedure A d d _ lis t(d im ,L L 2 ,L L l) relies on A_dd(dim,L2,LLl), which adds a single list L2 (i.e. a single m-function) to a list of lists LL1 (i.e. a list of m-functions), with the following outcome: if the list to be added does not exist in the list LL1 then the list LL1 is augmented by L2. I f on the other hand L2 is already present in the list LL1, then its multiplicity is increased by the multiplicity of L2. Unit 2 contains procedures to perform the expansion of a single m-function in terms of S-functions, S_f un c tio n (dim , c,L), and to express a list of m-functions in terms of S-functions, Schur_list(dim ,LL). The algorithm followed is the one explained in section B.2.2. The call command S_f unction (dim, c,L) has as input dim, the desired number of indeterminates as before, L the list specifying the m-function and c, a real number, the coefficient associated to the m-function. The zero parts of the input list may be omitted. For example the two following commands have identical output: > S_function(3,1,[3,0,0]); [[3, 0, 0, 1], [2, 1, 0, -1], [1, 1, 1, 1]] > S_function(3,l,[3]); [[3, 0, 0, 1], [2, 1, 0, -1], [1 , 1, 1, 1]] In other words, m {3) = S(3) - S(2 1 ) + S( 111 ) (B.46) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 160 On the other hand, the list-elements of the list LL in Schur_list(dim,LL) are ex pected to have dim + 1 parts, where the last one specifies the multiplicity of the m- function. This format is more convenient for use in conjunction with procedures described further down and designed to obtain the outer product, skew division and plethysm of S-functions. Clearly the procedures S_function and A dd-list are required. Unit 3 gathers together the procedures that construct the expansion of an S-function in terms of p-functions following equation (B.5). First, pow_sum(S) calls on ordena(S) to rearrange the input list S whose parts are in ascending order and return a list with the same parts but in descending order. This is necessary so that the library command that gives the character tables of the symmetric groups can be called. Further, to calculate the coefficient associated w ith each pp, the procedure pow_sum calls on orderc (L), which determines the order zp of the class identified by the partition L; i.e. evaluates equation (B.6 ). For example, > pow_sum([3]); [[1, 1, 1, 1/6], [1, 2, 1/2], [3, 1/3]] That is, 1 1 1 S(s) = g P( 13) + g P(21) + 3 P(3) (B.47) The procedures in unit 4 execute the algorithm detailed in section B.3.1 for the product and division of m-functions. Firstly, m_prod(dim,Ll ,L2) and m_div(dim,Ll ,L2) resolve the product and division of two m-functions. These rely on three more procedures: order_part, which puts in descending order the parts of a partition given in random order; equal_number_part, which checks whether two partitions are the same; and mdim, which evaluates the appropriate coefficients through (B.24). For example, > m_prod(4,[4,2,1,0,1],[2,1,0,0,1]); [[ 6 , 3, 1, 0, 1], [6 , 2, 2, 0, 2], [6 , 2, 1, 1, 2], [5, 4, 1, 0, 1], [5, 3, 2, 0, 1], [5, 2, 2, 1, 2], [4, 4, 2, 0, 2], [4, 4, 1, 1, 4], [4, 3, 3, 0, 2], [4, 3, 2, 1, 2], [4, 2, 2, 2, 6 ]] > m_div(4,[2,1,1,0,1] ,[1,1,1,0,1]); [[1, 0, 0, 0, 3], [1, 1, 0, -1 , 2], [2, 0, 0, -1, 1]] That is, in 4 parts or fewer, 771(421)771(21) = 7"(631) + 2771(622) + 2?77(6211) + ^(541) + ^(532) + 2777(5221) + 2777(442) + 4777(44n) + 2777(433) + 2777(4321) + 6777(4222)(B.48) and, in 4 parts or fewer, ^(211)/" H ill) = 3777(1000) + 2777(H0-1) + 777(200-1) (B.49) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 161 The multiplication and division of a list of m-functions by a list of m-functions is car ried out by M_prod(dim,LLl ,LL2) and M_div(dim,LLl,LL2). Once again the procedures Add_list and A_dd are required to add lists of m-functions. The routines that carry out the outer product and skew division of S-functions are found in unit 5. As explained in section B.3.2, the input S-functions are first expressed in terms of m-functions, which are then subjected to the multiplication or division routines of unit 4, and finally the resulting expansion converted back to S-functions. Clearly the execution of the commands in this unit requires that units 1, 3 and 4 be previously read. The call commands are outer_prod(dim, S1,S2) and outer_div(dim, S1,S2), where SI and S2 are simple lists specifying single S-functions. An example is > outer_prod(3,[3,2,1],[1]); [[3, 2, 2, 1], [4, 2, 1, 1], [3, 3, 1, 1]] > outer_div(6,[3,2,2,1,1,1],[1,1,1,1,1]); [[2, 1, 1, 1, 0, 0, 1], [2, 2, 1, 0, 0, 0, 1], [3, 1, 1, 0, 0, 0, 1]] That is, in no more than 3 parts, 5(321)5(1) = 5(421) + 5(331) + 5(322) (B.50) and, with no restriction on the number of parts in the result, 5(322111)/^(lll 11) = 5(311) + 5(221) + 5(2111) (B.51) Note that if a part of the S-function divisor is larger then the corresponding part of the S-function being divided the message “nuW appears. For example: > outer_div(4,[3,2,1],[1,1,1,1]); "null" > outer_div(4,[3,2,1],[3,3]); "null" It should also be noted here that entering as input a value for dim greater than the number of non-zero parts of the dividend is a waste of time. Unit 6 contains a single procedure inter_prod(Sl,S2) that performs the internal product of two S-functions. This procedure follows the algorithm described in section B.3.3 and makes substantial use of the MAPLE library combinat. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 162 > inter_prod([2,1],[2,1]); [C3, 1] , [2, 1, 1], [1, 1, 1, 1]] That is, S(21) O S(21) = 3) + S(21) + s(lll) (B.52) Finally, the procedures to calculate the outer plethysm of S-functions are found in unit 7. Units 1, 2, 3, and 4 are required to be read prior to executing this unit. The main procedure is plethysm (dim, SI ,S2) where as before SI and S2 stand for single S-functions. This routine calls first on the procedures that convert the S-function SI into m-functions (unit 1) and the S-function S 2 into p-functions (unit 3). Next, the procedure pie evaluates the plethysm of a list of monomial functions by a simple p- function (cf. (B.4)), the procedure plet evaluates the plethysm of a list of monomial functions by a product of p-functions (requires M_prod of unit 4) and the procedure p le th evaluates the plethysm of a list of monomial functions by a list of products of p-functions and requires Add_list. The procedure p le t also calls on a procedure C_prod(c,L), which multiplies the last element of a list L by a real coefficient c. To obtain the final result in S-functions unit 2 procedures are needed. An example is plethysm(3, [1 ,1 ] , [2]); [[2, 2, 0, 1]] That is, in 3 parts or fewer, S(11) © s(2) = S(22) (B.53) The generalization of the plethysm routine to a sum of S-functions on the left can be easily implemented with the following procedure: > Mplethysm:=proc(dim::nonnegint,SSl::list(list),S2::list)local S2p,Slm, SR; > Slm:=M_content(dim,SSl); > S2p:=pow_sum(S2); > SR:=pleth(dim,Slm,S2p);RETURN(Schur_list(dim,SR)); > end: The call command is Mplethysm(dim,SSl,S2) where SSI is a list of lists and S2 a single list. For example, > Mplethysm(3,[[2],[1,1]], [2]); [[2, 2, 0, 2], [3, 1, 0, 1], [4, 0, 0, 1], [2, 1, 1, 1]] That is, in 3 parts or fewer, (S(2) + S(n)) © S(2) = S(4) + S(3i) + 2S(22) + S(211) (B.54) It would be a simple matter to modify the program so that the list S2 could also be a sum of S-functions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 163 Acknowledgements This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. References • [1] D.E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, 2 edn., Clarendon Press, Oxford, 1950. • [2] S C H U R ™ , An interactive program for calculating properties of Lie groups and symmetric functions, distributed by S.M. Christensen, P.O. Box 16175, Chapel Hill, NC 27516; E-mail [email protected]; http://smc.vnet/Christensen.html • [3] M. J. Carvalho, S. D ’Agostino, Plethysms of Schur functions and the shell model, J. Phys. A: Math. Gen. 34 (2001) 1375-1392. • [4] I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2 edn, Clarendon Press, Oxford, 1995. • [5] R.A. Proctor, Young tableaux, Gelfand patterns, and branching rules for clas sical groups, J. Algebra 164 (1994) 299-360. • [6 ] M.J. Carvalho, S-function series revisited, J. Phys. A: Math. Gen. 26 (1993) 2179-2197. • [7] T. Muir, The Theory of Determinants, Vol. 4, Dover, New York, 1923 (reprinted 1960). • [8] R.C. King, Branching rules for classical Lie groups using tensor and spinor methods, J. Phys. A: Math. Gen. 8 (1975) 429-449. • [9] B.G. Wybourne, Symmetry Principles and Atomic Spectroscopy, Wiley-Interscience, New York, 1970. • [10] C.J. Cummins, Young Diagrams and Branching Rules, publication CRM 1471, Centre de Recherches Mathematiques, Universite de Montreal, Montreal, 1987. • [11] D.E. Littlewood, A.R. Richardson, Group characters and algebra, Roy. Soc. Lon. Philos. Trans. A233 (1934) 99-141 • [12] Y.M . Chen, A.M . Garsia, J. Remmel, Algorithm s for plethysms, in: C. Greene (Ed.), Combinatorics and Algebra, Contemporary Mathematics, Vol. 34, American Mathematical Society, Providence, 1984, pp. 59-84. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 164 TEST RUN INPUT and OUTPUT The run times of the following tests have been obtained w ith a laptop Compaq Pre- sario 1260 and w ith desktop computers having Pentium processors w ith speeds of 933 MHz and 300 MHz. Run times are summarized in a table placed at the end of the tests. • [Test 1] > outer_prod(3,[5,4,3],[4,3]); [[ 8 , 7, 4, 2], [9, 6 , 4, 1], [9, 7, 3, 1], [ 8 , 8 , 3, 1], [ 8 , 6 , 5, 1], [7, 7, 5, 1]] • [Test 2] > outer_prod(4,[5,4,3],[4,3]); [[ 6 , 5, 4, 4, 1], [7, 6 , 4, 2, 3], [ 8 , 5, 4, 2, 3], [ 8 , 6 , 3, 2, 2], [7, 7, 3, 2, 1], [7, 6 , 5, 1 , 2 , [ 8 , 6 , 4, 1, 3], [8, 4, 4, 3, 2], [ 6 , 6 , 4, 3, 1 , [ 6 , 5, 5, 3, 1], [8, 7, 4, 0, 2], [9, 4, 3, 3, 1 , [7, 5, 4, 3, 3], [7, 4, 4, 4, 1], [9, 4, 4, 2 , 1 , [7, 6 , 3, 3, 1], [7, 5, 5, 2, 2], [9, 6 , 3, 1 , 1 , [ 8 , 5, 3, 3, 2], [7, 7, 4, 1, 2], [9, 6 , 4, 0 , 1 , [9, 5, 3, 2, 1], [8, 7, 3, 1, 2], [ 8 , 5, 5, 1 , 1 , [9, 7, 3, 0, 1], [9, 5, 4, 1, 1], [ 8 , 8 , 3, 0 , 1 , [ 8 , 6 , 5, 0, 1], [7, 7, 5, 0, 1] [ 6 , 6 , 5, 2 , 1 ] • [Test 3] > outer_prod(5,[5,4,3],[4,3]); CO , 4 : , 3 , 1], [5, 4, 4, 4, 2 , 1] 9 [7 , 4, 4, 2, 2, 1] 9 [ 6 , 5, 4, 3, 1, 3] , [ 6 , 4, 4, 4, 1 , 1 ], [7, 5, 4, 2 , 1 , 3 ], [ 6 , 5, 4, 4, 0, 1] , [ 8 , 5, 3, 2 , 1 , 1 ], [7 , 5, 5, 1 , 1 , 1], [7, 6 , 4, 1 , 1 , 2 ], [7, 6 , 4, 2 , 0 , 3 ], [ 8 , 5, 4, 2 , o, 3 ], 1 __ 00 1 6 , 3, 2 , 0 , 2 ], [7, 7, 3, 2 , 0 , 1] , [ 6 , 5, 4, 2 , 2 , 2], [7, 6 , 5, 1 , o, 2] , [ 8 , 6 , 4, 1 , 0 , 3 ], [ 8 , 4, 4, 3, 0 , 2], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 165 [5, 5, 4, 3, 2, 2] [6, 6, 4, 3, 0, 1], [6, 5, 5, 3, 0, 1], [8, 7, 4, 0, 0, 2] [6, 5, 5, 2, 1, 2], [7, 4, 4, 3, 1, 2], [7, 5, 3, 3, 1, 2] [8, 6, 3, 1, 1, 1], [9, 4, 3, 3, 0, 1], [7, 5, 4, 3, 0, 3] [5, 5, 4, 4, 1, 1], [7, 4, 4, 4, 0, 1], [9, 4, 4, 2, 0, 1] [7, 6, 3, 3, 0, 1], [7, 5, 5, 2, 0, 2], [5, 5, 5, 3, 1, 1] [6, 6, 3, 2, 2, 1], [8, 5, 4, 1, 1, 1], [9, 6, 3, 1, 0, 1] [8, 5, 3, 3, 0, 2], [7, 7, 3, 1, 1, 1], [7, 5, 3, 2, 2, 1] [6, 6, 3, 3, 1, 1], [7, 6, 3, 2, 1, 2], [8, 4, 4, 2, 1, 1] [7, 7, 4, 1, 0, 2], [6, 4, 4, 3, 2, 2], [7, 4, 3, 3, 2, 1] [9, 6, 4, 0, 0, 1], [9, 5, 3, 2, 0, 1], [8, 7, 3, 1, 0, 2] [6, 6, 4, 2, 1, 2], [8, 5, 5, 1, 0, 1], [8, 4, 3, 3, 1, 1] [6, 5, 3, 3, 2, 2], [9, 7, 3, 0, 0, 1], [9, 5, 4, 1, 0, 1] [8, 8, 3, 0, 0, 1], [8, 6, 5, 0, 0, 1], [7, 7, 5, 0, 0, 1] [6, 6, 5, 1, 1, 1], [6, 6, 5, 2, 0, 1], [5, 5, 5, 2, 2, 1] [6, 4, 3, 3, 3, 1], [5, 5, 3, 3, 3, 1]] • [Test 4] > outer_div(4,[5,4,3,2], [2,1]); [[3, 3, 3, 2, 1], [4, 4, 2, 1, 2], [4, 4, 3, 0, 1], [5, 3, 2, 1, [5, 3, 3, 0, 1], [5, 4, 1, 1, 1], [5, 2, 2, 2, 1], [4, 3, 2, 2, 2], [5, 4, 2, 0, 1], [4, 3, 3, 1, 2]] • [Test 5] > inter_prod([5,4,3], [10,1,1]); [[7, 3, 2, 1], [6, 5, 1, 1], [6, 4, 2, 3], [6, 4, 1, 1, 1], [6, 3, 3, 2], [6, 3, 2, 1, 2], [5, 5, 2, 2], [5, 5, 1, 1, 1], [5, 4, 3, 4], [5, 4, 2, 1, 4], [5, 4, 1, 1, 1, 1], [5, 3, 3,1, 3], [5, 3, 2,2, 1], [5, 3, 2, 1, 1, 1], [4, 4, 4, 1], [4, 4, 3, 1, 3], [4, 4, 2, 2, 1], [4, 4, 2, 1, 1, 1], [4, 3, 3, 2, 1], [4, 3, 3, 1, 1, 1]] • [Test 6 ] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B . 166 > plethysm(3, [3,2,2], [2,2]); [[ 1 0 , 1 0 , 8 , 1]] • [Test 7] > plethysm(4,[3,2,2], [2,2]); [[ 8 , 8 , 8 , 4, 8], [8 , 8 , 7, 5, 9], [ 8 , 7, 7, 6 , 5], [7, 7, 7, 7, 2], [8 , 8 , 6 , 6 , 10], [9, 8 , 7, 4, 20], [9, 8 , 6 , 5, 19], [9, 7, 7, 5, 14], [9, 7, 6 , 6 , 8], [9, 8 , 8 , 3, 8], [10, 8 , 6 , 4, 19], [10, 8 , 5, 5, 5], [10, 7, 7, 4, 8], [10, 6 , 6 , 6 , 4 ], [9, 9, 6 , 4, 9], [9, 9, 5, 5, 8], [10, 7, 6 , 5, 12], [9, 9, 7, 3, 11], [9, 9, 8 , 2, 3], [11, 8 , 5, 4, 4 ], [11, 7, 6 , 4, 5], [11, 6 , 6 , 5, 2 ], [11, 7, 5, 5, 3], [11, 8 , 6 , 3, 5 ], [10, 9, 6 , 3, 8], [11, 8 , 7, 2, 3], [10, 9, 7, 2, 5], [11, 8 , 8 , 1, 1 ], [10, 9, 8 , 1, 2], [12, 8 , 4, 4, 1], [10, 10, 4, 4, 2], [12, 7, 6 , 3, 1], [11, 9, 5, 3, 2], [10, 8 , 7, 3, 11], [12, 8 , 6 , 2, 1 ], [11, 9, 6 , 2, 1], [11, 7, 7, 3, 3], [10, 8 , 8 , 2, 6 ] , [10, 10, 6 , 2, 2], [10, 9, 5, 4, 6 ], [12, 6 , 6 , 4, 1], [11, 9, 7, 1, 1], [9, 9, 9, 1, 1], [1 0 , 1 0 , 8 , 0 , 1]] • [Test 8] > Mplethysm(3,[[1 ],[2 ], [3]], [2,1]); [[4, 2, 2, 1], [3, 2, 2 , 1], [5, 3, 1 , 1], [4, 3, 1 , 2] y [5, 2 , 1 , 2], [4, 2 , 1, 3], [3, 3, 1 , 2], [3, 2 , 1 , 3], [2 , 2 , 1 , 1], [6 , 3, 0, 1], [5, 4, 0, 1], [4, 3, 2 , 1], [7, 2 , 0, 1], [5, 3, 0, 3], [4, 4, 0, 1], [3, 3, 2 , 1], [6 , 2 , 0, 3], [5, 2 , 0, 5], [4, 3, 0, 4], [8 , 1 , 0, 1], [6 , 2 , 1 , 1], [7, 1 , 0 , 2], [6 , 1 , 0, 4], [4, 2 , 0, 5], [3, 3, 0 , 2], [5, 1 , 0, 5], [4, 1 , 0, 4], [3, 2 , 0, 3], [8 , 0, 0, 1], [6 , 1 , 1 , 1], [7, 0, 0 , 2], [5, 1 , 1 , 2], [6 , 0, 0 , 2], [4, 1 , 1 , 2], [5, 0, 0 , 2], [3, 1 , 1 , 2], [3, 1 , 0 , 2], [2 , 2 , 0, 1], [4, 0, 0, 1],[2 , 1 , 1 , 1], [2 , 1 , 0, 1]] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p e n d ix B. 167 Table 1. Test run times (all times are in seconds). Presario 1260 Pentium 300 Pentium 933 Test 1 0.4 0 . 0 2 Test 2 7.9 7.9 1.4 Test 3 236.0 120.3 22.4 Test 4 1.5 0.4 0 . 2 Test 5 17.3 6.9 2.9 Test 6 Test 7 30.0 18.2 3.5 Test 8 8.5 5.5 1 . 2 Blank spaces in the table indicate that the operation was completed so quickly that M APLE’s counter did not register any elapsed processor time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (u) = (A) -I (3.7) For the case of one-row irreps of U(ran), i.e., where A = (N), it is known from the character theory of symmetric groups that the only partitions that satisfy the restriction in equation (3.7) are those for which ji — v. For the case of one-column irreps of U(mn), i.e., where A = (1N), it is known from the character theory of symmetric groups that the only partitions that satisfy the restriction in equation (3.7) are those for which /i = v. Thus, both cases of unitary-unitary duality follow from Schur-Weyl duality. [b © {*/}] In passing from the penultimate line to the last line of the previous diagram, one forms