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 ..................................................................