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Representing finite groups: a semisimple introduction

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Bibliographic Details
Main Author:	Sengupta, Ambar 1963- (Author)
Format:	Book
Language:	English
Published:	New York, NY [u.a.] Springer 2012
Subjects:	Darstellungstheorie Endliche Gruppe
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027202226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	XV, 371 S.
ISBN:	9781461412304

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Record in the Search Index

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adam_text	Contents Concepts and Constructs 1 1.1 Representations of Groups ................... 2 1.2 Representations and Their Morphisms ............. 4 1.3 Direct Sums and Tensor Products ............... 4 1.4 Change of Field ......................... 5 1.5 Invariant Subspaces and Quotients ............... (i 1.6 Dual Representations ...................... 7 1.7 Irreducible Representations ................... 10 1.8 Schürt Lemma .......................... 12 1.9 The Frobonius Schur Indicator ................. 14 1.10 Character of a Representation ................. 1С 1.11 Diagonalizability ......................... 20 1.12 Unitarity ............................. 22 1.13 Rival Reads ........................... 24 1.14 Afterthoughts: Lattices ..................... 25 Exercises .............................. 28 A Reckoning ............................... 37 Basic Examples 39 2.1 Cyclic Groups .......................... 40 2.2 Dihedral Groups ......................... 43 2.3 The Symmetric Group 54 .................... 48 2.4 Quaternionic Units ....................... 52 2.5 Afterthoughts: Geometric Groups ............... 54 Exercises .............................. 56 XI XII CONTENTS 3 The Group Algebra 59 3.1 Definition of the Group Algebra ................ 60 3.2 Representations of G and ¥[G] ................. 61 3.3 The Center ............................ 63 3.4 Deconstructing ¥[S3] ...................... 65 3.5 When ¥[G) Is Semisimple.................... 73 3.6 Afterthoughts: Invariants .................... 77 Exercises .............................. 79 4 More Group Algebra 83 4.1 Looking Ahead .......................... 84 4.2 Submodules and Idempotents .................. 86 4.3 Deconstructing ¥ G), the Module ................ 89 4.4 Deconstructing ¥[G], the Algebra ............... 91 4.5 As Simple as Matrix Algebras ................. 97 4.6 Putting ¥[G] Back Together ..................102 4.7 The Mother of All Representations ...............107 4.8 The Center ...........................109 4.9 Representing Abelian Groups ..................111 4.10 Indecomposable Idempotents ..................112 4.11 Beyond Our Borders .......................114 Exercises ..............................115 5 Simply Semisimple 125 5.1 Schur s Lemma ..........................126 5.2 Semisimple Modules .......................127 5.3 Deconstructing Semisimple Modules ..............131 5.4 Simple Modules for Semisimple Rings .............134 5.5 Deconstructing Semisimple Rings ...............136 5.6 Simply Simple ..........................140 5.7 Commutants and Double Commutants .............141 5.8 Artin-Wedderburn Structure ..................144 5.9 A Module as the Sum of Its Parts ...............145 510 Readings on Rings ........................147 5.11 Afterthoughts: Clifford Algebras ................147 Exercises ..............................151 CONTENTS ХШ 6 Representations of Sn 157 6.1 Permutations and Partitions .................. 157 6.2 Complements and Young Tableaux ............... 161 6.3 Symmetries of Partitions ....................165 6.4 Conjugacy Classes to Young Tableaux ............. 168 6.5 Young Tableaux to Young Symmetrizers ............ 169 6.6 Youngtabs to Irreducible Representations ........... 170 6.7 Youngtab Apps ......................... 174 6.8 Orthogonality .......................... 179 6.9 Deconstructing ¥[Sn] ...................... 180 6.10 Integrality ............................ 183 6.11 Rivals and Rebels ........................ 184 6.12 Afterthoughts: Reflections ................... 184 Exercises .............................. 188 7 Characters 189 7.1 The Regular Character ....................190 7.2 Character Orthogonality ....................194 7.3 Character Expansions ......................203 7.4 Comparing Z-Bases .......................206 7.5 Character Arithmetic ......................208 7.6 Computing Characters .....................211 7.7 Return of the Group Determinant ........-......214 7.8 Orthogonality of Matrix Elements ..............217 7.9 Solving Equations in Groups ..................219 7.10 Character References ......................228 7.11 Afterthoughts: Connections .......-...........229 Exercises ..............................230 8 Induced Representations 235 8.1 Constructions ..........................235 8.2 The Induced Character .....................238 8.3 Induction Workout ........................239 8.4 Universality ...........................242 8.5 Universal Consequences .....................243 8.6 Reciprocity ............................245 8.7 Afterthoughts: Numbers ....................247 Exercises ..............................248 XIV CONTENTS 9 Commutant Duality 249 9-1 The Commutant .........................249 9.2 The Double Commutant .................... 251 9.3 Commutant Decomposition of a Module ............254 9.4 The Matrix Version .......................260 Exercises ..............................264 10 Character Duality 267 10.1 The Commutant for 5n on V®n ................267 10.2 Schur-Weyl Duality .......................269 10.3 Character Duality, the High Road ...............270 10.4 Character Duality by Calculations ...............271 Exercises ..............................278 11 Representations of ГУ (TV) 281 11.1 The Haar Integral ........................ 282 11.2 The Weyl Integration Formula ................. 283 11.3 Character Orthogonality ................... 284 11.4 Weights .......................... 285 11.5 Characters of U{N) ...................... 286 11.6 Weyl Dimension Formula .................... 290 11.7 From Weights to Representations ................ 291 11.8 Characters of Sn from Characters of U(N) ...........294 Exercises ..............................299 12 Postscript: Algebra 301 12.1 Groups and Less .........................301 12.2 Rings and More .........................306 12.3 Fields ...............................314 12.4 Modules over Rings ........................315 12.5 Free Modules and Bases .....................319 12.6 Power Series and Polynomials ..................323 12.7 Algebraic Integers .........................329 12.8 Linear Algebra ..........................330 12.9 Tensor Products .........................335 12.10 Extension of Base Ring .......-.............338 CONTENTS XV 12.11 Determinants and TV ас es of Matrices ............. 339 12.12 Exterior Powers ......................... 340 12.13 Eigenvalues and Eigenvectors .................. 345 12.14 Topology, Integration, and Hilbeit Spaces ........... 346 Bibliography 353 Index 361
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spellingShingle	Sengupta, Ambar 1963- Representing finite groups a semisimple introduction Darstellungstheorie (DE-588)4148816-7 gnd Endliche Gruppe (DE-588)4014651-0 gnd
subject_GND	(DE-588)4148816-7 (DE-588)4014651-0
title	Representing finite groups a semisimple introduction
title_auth	Representing finite groups a semisimple introduction
title_exact_search	Representing finite groups a semisimple introduction
title_full	Representing finite groups a semisimple introduction Ambar N. Sengupta
title_fullStr	Representing finite groups a semisimple introduction Ambar N. Sengupta
title_full_unstemmed	Representing finite groups a semisimple introduction Ambar N. Sengupta
title_short	Representing finite groups
title_sort	representing finite groups a semisimple introduction
title_sub	a semisimple introduction
topic	Darstellungstheorie (DE-588)4148816-7 gnd Endliche Gruppe (DE-588)4014651-0 gnd
topic_facet	Darstellungstheorie Endliche Gruppe
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027202226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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