DOCSLIB.ORG

By Santo D'agostino a Thesis Submitted in Conformity with The

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
By Santo D'agostino a Thesis Submitted in Conformity with The 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. ii 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. iii 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. iv 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 v 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 ..................................................................
Load more

Recommended publications
  • Pre-Publication Accepted Manuscript
    Sami H. Assaf, David E. Speyer Specht modules decompose as alternating sums of restrictions of Schur modules Proceedings of the American Mathematical Society DOI: 10.1090/proc/14815 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9939(XX)0000-0 SPECHT MODULES DECOMPOSE AS ALTERNATING SUMS OF RESTRICTIONS OF SCHUR MODULES SAMI H. ASSAF AND DAVID E. SPEYER (Communicated by Benjamin Brubaker) Abstract. Schur modules give the irreducible polynomial representations of the general linear group GLt. Viewing the symmetric group St as a subgroup of GLt, we may restrict Schur modules to St and decompose the result into a direct sum of Specht modules, the irreducible representations of St. We give an equivariant M¨obiusinversion formula that we use to invert this expansion in the representation ring for St for t large.
    [Show full text]
  • Boolean Product Polynomials, Schur Positivity, and Chern Plethysm
    BOOLEAN PRODUCT POLYNOMIALS, SCHUR POSITIVITY, AND CHERN PLETHYSM SARA C. BILLEY, BRENDON RHOADES, AND VASU TEWARI Abstract. Let k ≤ n be positive integers, and let Xn = (x1; : : : ; xn) be a list of n variables. P The Boolean product polynomial Bn;k(Xn) is the product of the linear forms i2S xi where S ranges over all k-element subsets of f1; 2; : : : ; ng. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials Bn;k(Xn) for certain k to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate Bn;n−1(Xn) to a bigraded action of the symmetric group Sn on a divergence free quotient of superspace. 1. Introduction The symmetric group Sn of permutations of [n] := f1; 2; : : : ; ng acts on the polynomial ring C[Xn] := C[x1; : : : ; xn] by variable permutation. Elements of the invariant subring Sn (1.1) C[Xn] := fF (Xn) 2 C[Xn]: w:F (Xn) = F (Xn) for all w 2 Sn g are called symmetric polynomials. Symmetric polynomials are typically defined using sums of products of the variables x1; : : : ; xn. Examples include the power sum, the elementary symmetric polynomial, and the homogeneous symmetric polynomial which are (respectively) (1.2) k k X X pk(Xn) = x1 + ··· + xn; ek(Xn) = xi1 ··· xik ; hk(Xn) = xi1 ··· xik : 1≤i1<···<ik≤n 1≤i1≤···≤ik≤n Given a partition λ = (λ1 ≥ · · · ≥ λk > 0) with k ≤ n parts, we have the monomial symmetric polynomial X (1.3) m (X ) = xλ1 ··· xλk ; λ n i1 ik i1; : : : ; ik distinct as well as the Schur polynomial sλ(Xn) whose definition is recalled in Section 2.
    [Show full text]
  • THE COMPUTATIONAL COMPLEXITY of PLETHYSM COEFFICIENTS Nick Fischer and Christian Ikenmeyer
    comput. complex. (2020) 29:8 c The Author(s) 2020 1016-3328/20/020001-43 published online November 4, 2020 https://doi.org/10.1007/s00037-020-00198-4 computational complexity THE COMPUTATIONAL COMPLEXITY OF PLETHYSM COEFFICIENTS Nick Fischer and Christian Ikenmeyer Abstract. In two papers, B¨urgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker co- efficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity. In this paper, we show that deciding positivity of plethysm coefficients is NP-hard and that computing plethysm coefficients is #P-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we con- sider to be of independent interest.
    [Show full text]
  • Ribbon Tableaux, Hall-Littlewood Functions and Unipotent Varieties ∗
    RIBBON TABLEAUX, HALL-LITTLEWOOD FUNCTIONS AND UNIPOTENT VARIETIES ∗ Alain Lascouxy, Bernard Leclercy and Jean-Yves Thibonz Abstract We introduce a new family of symmetric functions, which are defined in terms of ribbon tableaux and generalize Hall-Littlewood functions. We present a series of conjectures, and prove them in two special cases. 1 Introduction Hall-Littlewood functions [Li1] are known to be related to a variety of topics in rep- resentation theory, geometry and combinatorics. These symmetric functions arise in the character theory of finite linear groups [Gr], in the geometry of unipotent varieties [Sh1, HSh], in particular as characteristics of the representations of the symmetric group in their cohomology [HS], and appear to be related to the Quan- tum Inverse Scattering Method [KR]. From a combinatorial point of view, their description involves the deepest aspects of the theory of Young tableaux: the mul- tiplicative structure (plactic monoid) and the ordered structure derived from the cyclage operation [LS2, La]. There exists also a description in terms of Kashiwara's theory of crystal bases [LLT4]. Another kind of application of Hall-Littlewood functions is concerned with the representation theory of the complex linear group GL(n; C). The general setting is the following. Suppose we are given a finite dimensional representation V of G = GL(n; C). The symmetric group Sk acts (on the right) on the tensor space k W = V ⊗ by v1 v2 vk σ = vσ(1) vσ(2) vσ(k) : ⊗ ⊗ · · · ⊗ · ⊗ ⊗ · · · ⊗ r Let γ Sk be a k-cycle. The eigenvalues of γ, as an endomorphism of W are ζ , r = 0; :2 : : ; k 1, where ζ is a primitive k-th root of unity.
    [Show full text]
  • Plethysm and Lattice Point Counting 3
    PLETHYSM AND LATTICE POINT COUNTING THOMAS KAHLE AND MATEUSZ MICHALEK Abstract. We apply lattice point counting methods to compute the multiplicities in the plethysm of GL(n). Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition µ of 3, 4, or 5 we obtain an explicit formula in λ and k for the multiplicity of Sλ in Sµ(Sk). Contents 1. Introduction 1 2. Characters 6 3. Reductions 8 4. Asymptotic behavior 13 5. Appendix 16 References 23 1. Introduction The plethysm problem can be stated in different ways. One is to describe the homo- geneous polynomials on the spaces SkW ∗ and k W ∗ in terms of representations of the group GL(W ). This is equivalent to decomposing Sd(SkW ) into isotypic components and finding the multiplicity of each isotypic component.V The general goal in plethysm is to determine the coefficients of Sλ in Sµ(SνW ) as a function of the partitions λ,µ, and ν. The term plethysm was coined by Littlewood [Lit36], and this type of problems appears in many branches of mathematics beyond representation theory (consult [LR11] for some arXiv:1408.5708v3 [math.RT] 12 Aug 2015 recent developments in plethystic calculus). A general explicit solution of plethysm may be intractable as the resulting formulas are simply too complicated. Here we show piecewise quasi-polynomial formulas that describe the plethysm and then focus on two directions. One is explicit descriptions for small µ which we find with the help of computer algebra.
    [Show full text]
  • Representations of Reductive Lie Groups
    Representations of reductive Lie groups Lectures delivered by Joe Harris Notes by Akhil Mathew Spring 2013, Harvard Contents Lecture 1 1/28 x1 Mechanics 5 x2 Philosophy 6 x3 Basic definitions 7 x4 Examples 9 Lecture 2 1/30 x1 Examples 11 x2 More examples 12 x3 General remarks 13 x4 Neighborhoods of e generate 14 x5 Isogenies and covering spaces 15 Lecture 3 2/1 x1 Recap 16 x2 Isogeny 18 x3 The adjoint representation 19 x4 Differentiating the adjoint representation 21 Lecture 4 2/4 x1 The basic setup 22 x2 Describing the bracket 23 x3 Some general re- marks 25 x4 Lie brackets and commutators 26 x5 Some more terminology 26 x6 Representations of Lie algebras 27 Lecture 5 2/6 x1 Recap 28 x2 The exponential map 31 Lecture 6 2/8 x1 The exponential map 33 x2 The Baker-Campbell-Hausdorff formula 35 Lecture 7 2/11 x1 The dictionary 38 x2 Nilpotent, solvable, and semisimple Lie algebras 39 x3 Engel's and Lie's theorems 41 Lecture 8 2/13 x1 Engel's theorem 42 x2 Lie's theorem 45 1 Lecture 9 2/15 x1 The radical 47 x2 Jordan decomposition 49 x3 An example: sl2 51 Lecture 10 2/20 x1 sl2(C) 53 x2 Irreducible representations 54 Lecture 11 2/22 x1 Recap 58 x2 Plethysm 59 x3 sl3 60 Lecture 12 2/25 x1 Recap on sl3 63 x2 Irreducible representations of sl3 64 Lecture 13 2/27 x1 Continuation of sl3 67 x2 Irreducible representations 70 Lecture 14 3/1 x1 sl3 71 x2 Examples 73 Lecture 15 3/4 x1 Examples 77 Lecture 16 3/6 x1 Outline 78 Lecture 17 3/8 Lecture 18 3/11 x1 The Killing form 85 x2 sln 87 Lecture 19 3/13 x1 sln 90 Lecture 20 3/15 x1 Geometric plethysm 92
    [Show full text]
  • On Enumeration in Classical Invariant Theory 1
    ON ENUMERATION IN CLASSICAL INVARIANT THEORY BRUCE W. WESTBURY 1. Introduction For Cayley and Sylvester the central problem of invariant theory was to deter- mine the structure of the ring of invariants of a quantic. In the language of repre- sentation theory, the problem is to determine the structure of the ring of invariant polynomials on the symmetric power of a complex vector space. A preliminary problem is to determine the Hilbert series of the ring of invariant polynomials. Fol- lowing the development of quantum mechanics, attention moved on to invariant tensors. The new feature is that the symmetric group acts naturally on tensor powers by permuting tensor indices. This leads to the general problem. Let G be a reductive complex algebraic group and V a finite dimensional rational represen- tation. Then each isotypic subspace of ⊗rV has a natural action of the symmetric group, Sr, and the problem is to determine the Frobenius characters of these repre- sentations. There a several cases of this problem that have been studied but there are remarkably few cases for which the problem has been solved. The introduc- tion of [7] states that the problem of determining the characters for the adjoint representation of a classical group is \certainly intractable". The main result of this paper is a contribution to finding these characters. As- sume we have a representation V for which the characters of the W -isotypic sub- spaces are known. Then Theorem 2 is a formula for the characters of the W -isotypic subspaces for any representation of the form P (V ) where P is a polynomial func- tor, for example a symmetric or exterior power.
    [Show full text]
  • Branching from the General Linear Group to the Symmetric Group and the Principal Embedding
    Branching from the General Linear Group to the Symmetric Group and the Principal Embedding Alexander Heaton, Songpon Sriwongsa, Jeb F. Willenbring May 12, 2020 Abstract Let S be a principally embedded sl2-subalgebra in sln for n ≥ 3. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible sln-representation, V , there exists an irreducible S-representation embedding in V with dimension at most b(n). In a 2017 paper (joint with Hassan Lhou), they prove that b(n) = n is the sharpest possible bound, and also address embeddings other than the principal one. These results concerning embeddings may by interpreted as statements about plethysm. Then, in turn, a well known result about these plethysms can be interpreted as a \branching rule". Specifically, a finite dimensional irreducible representation of GL(n; C) will decompose into irreducible representations of the symmetric group when it is restricted to the subgroup consisting of permutation matrices. The question of which irreducible representations of the symmetric group occur with positive multi- plicity is the topic of this paper, applying the previous work of Lhou, Zuckerman, and the third author. A complex irreducible representation V of sl2(C) defines a homomorphism arXiv:1812.06211v2 [math.RT] 9 May 2020 π : sl2 ! End(V ): ∼ Fixing an ordered basis we obtain an identification End(V ) = gln. Since sl2 is a simple Lie algebra, the kernel is trivial and the image of π, denoted s, is therefore isomorphic to sl2. We will refer to s as a principal sl2-subalgebra of gln.
    [Show full text]
  • Invariance Properties for Coefficients of Symmetric Functions
    Invariance properties for coefficients of symmetric functions Emmanuel Briand, Rosa Orellana, Mercedes Rosas To cite this version: Emmanuel Briand, Rosa Orellana, Mercedes Rosas. Invariance properties for coefficients of symmet- ric functions. 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), Jul 2015, Daejeon, South Korea. pp.619-630. hal-01337788 HAL Id: hal-01337788 https://hal.archives-ouvertes.fr/hal-01337788 Submitted on 27 Jun 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License FPSAC 2015, Daejeon, South Korea DMTCS proc. FPSAC’15, 2015, 619–630 Invariance properties for coefficients of symmetric functions Emmanuel Briand1y and Rosa Orellana 2zand Mercedes Rosas 3x 1Departamento de Matematica´ Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes, 41012 Sevilla, Spain. 2Dartmouth College, Mathematics Department, 6188 Kemeny Hall, Hanover, NH 03755, USA. 3Departamento de Algebra,´ Universidad de Sevilla, Avda. Reina Mercedes, 41012 Sevilla, Spain. Abstract. We show that several of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share invariance properties related to the operations of taking complements with respect to rectangles and adding rectangles.
    [Show full text]
  • A Computational and Combinatorial Expos้ of Plethystic Calculus
    J Algebr Comb (2011) 33: 163–198 DOI 10.1007/s10801-010-0238-4 A computational and combinatorial exposé of plethystic calculus Nicholas A. Loehr · Jeffrey B. Remmel Received: 29 January 2010 / Accepted: 19 May 2010 / Published online: 12 June 2010 © The Author(s) 2010. This article is published with open access at Springerlink.com Abstract In recent years, plethystic calculus has emerged as a powerful technical tool for studying symmetric polynomials. In particular, some striking recent advances in the theory of Macdonald polynomials have relied heavily on plethystic computa- tions. The main purpose of this article is to give a detailed explanation of a method for finding combinatorial interpretations of many commonly occurring plethystic ex- pressions, which utilizes expansions in terms of quasisymmetric functions. To aid newcomers to plethysm, we also provide a self-contained exposition of the funda- mental computational rules underlying plethystic calculus. Although these rules are well-known, their proofs can be difficult to extract from the literature. Our treatment emphasizes concrete calculations and the central role played by evaluation homomor- phisms arising from the universal mapping property for polynomial rings. Keywords Plethysm · Symmetric functions · Quasisymmetric functions · LLT polynomials · Macdonald polynomials 1 Introduction 1.1 Plethysm The plethysm F [G] of a symmetric polynomial F(x) with a symmetric polynomial G(x) is essentially the polynomial obtained by substituting the monomials of G(x) First author supported in part by National Security Agency grant H98230-08-1-0045. N.A. Loehr () Virginia Tech, Blacksburg, VA 24061-0123, USA e-mail: [email protected] J.B. Remmel University of California, San Diego, La Jolla CA 92093-0112, USA e-mail: [email protected] 164 J Algebr Comb (2011) 33: 163–198 for the variables of F(x).
    [Show full text]
  • Symmetric Functions
    7 Symmetric Functions 7.1 Symmetric Functions in General The theory of symmetric functions has many applications to enumerative combi- natorics, as well as to such other branches of mathematics as group theory, Lie algebras, and algebraic geometry. Our aim in this chapter is to develop the basic combinatorial properties of symmetric functions; the connections with algebra will only be hinted at in Sections 7.18 and 7.24, Appendix 2, and in some exercises. Let x = (x\, Jt2,...) be a set of indeterminates, and let n eN. A homogeneous symmetric function of degree n over a commutative ring R (with identity) is a formal power series where (a) a ranges over all weak compositions a = («i, «2, • • •) of n (of infinite a x length), (b) ca e R, (c) x stands for the monomial x\ x£ ..., and (d) /(x^i), Xw(2), • • •) = f(x\,X2,.. •) for every permutation w of the positive integers P. (A symmetric function of degree 0 is just an element of R.) Note that the term "symmetric function" is something of a misnomer; f(x) is not regarded as a function but rather as a formal power series. Nevertheless, for historical reasons we adhere to the above terminology. The set of all homogeneous symmetric functions of degree n over R is denoted n n A R. Clearly if /, g e A R and a, b e R, then af + bg e A\\ in other words, A^ is an R-module. For our purposes it will suffice to take R = Q (or sometimes Q with some indeterminates adjoined), so A^ is a Q-vector space.
    [Show full text]
  • The Computational Complexity of Plethysm Coefficients
    The Computational Complexity of Plethysm Coefficients Nick Fischer∗ Christian Ikenmeyery Max Planck Institute for Informatics, The University of Liverpool Saarbr¨ucken, Germany Saarbr¨ucken Graduate School of Computer Science Abstract In two papers, B¨urgisserand Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity. In this paper we show that deciding positivity of plethysm coefficients is NP-hard, and that computing plethysm coefficients is #P-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017).
    [Show full text]