Representing finite groups : a semisimple introduction

This graduate textbook presents the basics of representation theory for finite groups from the point of view of semisimple algebras and modules over them. The presentation interweaves insights from specific examples with development of general and powerful tools based on the notion of semisimplicity. The elegant ideas of commutant duality are introduced, along with an introduction to representations of unitary groups. The text progresses systematically and the presentation is friendly and inviting. Central concepts are revisited and explored from multiple viewpoints. Exercises at the end of the chapter help reinforce the material. __Representing Finite Groups: A Semisimple Introduction__ would serve as a textbook for graduate and some advanced undergraduate courses in mathematics. Prerequisites include acquaintance with elementary group theory and some familiarity with rings and modules. A final chapter presents a self-contained account of notions and results in algebra that are used. Researchers in mathematics and mathematical physics will also find this book useful. A separate solutions manual is available for instructors. Representing Finite Groups - A Semisimple Introduction 1 Title page 4 Preface 8 Contents 12 Chapter 1 Concepts and Constructs 18 1.1 Representations of Groups 19 1.2 Representations and Their Morphisms 21 1.3 Direct Sums and Tensor Products 21 1.4 Change of Field 22 1.5 Invariant Subspaces and Quotients 23 1.6 Dual Representations 24 1.7 Irreducible Representations 27 1.8 Schur's Lemma 29 1.9 The Frobenius–Schur Indicator 31 1.10 Character of a Representation 33 1.11 Diagonalizability 37 1.12 Unitarity 39 1.13 Rival Reads 41 1.14 Afterthoughts: Lattices 42 Exercises 45 A Reckoning 54 Chapter 2 Basic Examples 56 2.1 Cyclic Groups 57 2.2 Dihedral Groups 60 2.3 The Symmetric Group S4 65 2.4 Quaternionic Units 69 2.5 Afterthoughts: Geometric Groups 71 Exercises 73 Chapter 3 The Group Algebra 76 3.1 Definition of the Group Algebra 77 3.2 Representations of G and F[G] 78 3.3 The Center 80 3.4 Deconstructing F[S3] 82 3.5 When F[G] Is Semisimple 90 3.6 Afterthoughts: Invariants 94 Exercises 96 Chapter 4 More Group Algebra 99 4.1 Looking Ahead 100 4.2 Submodules and Idempotents 102 4.3 Deconstructing F[G], the Module 105 4.4 Deconstructing F[G], the Algebra 107 4.5 As Simple as Matrix Algebras 113 4.6 Putting F[G] Back Together 118 4.7 The Mother of All Representations 123 4.8 The Center 125 4.9 Representing Abelian Groups 127 4.10 Indecomposable Idempotents 128 4.11 Beyond Our Borders 130 Exercises 131 Chapter 5 Simply Semisimple 140 5.1 Schur's Lemma 141 5.2 Semisimple Modules 142 5.3 Deconstructing Semisimple Modules 146 5.4 Simple Modules for Semisimple Rings 149 5.5 Deconstructing Semisimple Rings 151 5.6 Simply Simple 155 5.7 Commutants and Double Commutants 156 5.8 Artin–Wedderburn Structure 159 5.9 A Module as the Sum of Its Parts 160 5.10 Readings on Rings 162 5.11 Afterthoughts: Clifford Algebras 162 Exercises 166 Chapter 6 Representations of Sn 172 6.1 Permutations and Partitions 172 6.2 Complements and Young Tableaux 176 6.3 Symmetries of Partitions 180 6.4 Conjugacy Classes to Young Tableaux 183 6.5 Young Tableaux to Young Symmetrizers 184 6.6 Youngtabs to Irreducible Representations 185 6.7 Youngtab Apps 189 6.8 Orthogonality 194 6.9 Deconstructing F[Sn] 195 6.10 Integrality 198 6.11 Rivals and Rebels 199 6.12 Afterthoughts: Reflections 199 Exercises 203 Chapter 7 Characters 204 7.1 The Regular Character 205 7.2 Character Orthogonality 209 7.3 Character Expansions 218 7.4 Comparing Z-Bases 221 7.5 Character Arithmetic 223 7.6 Computing Characters 226 7.7 Return of the Group Determinant 229 7.8 Orthogonality of Matrix Elements 232 7.9 Solving Equations in Groups 234 7.10 Character References 243 7.11 Afterthoughts: Connections 244 Exercises 245 Chapter 8 Induced Representations 250 8.1 Constructions 250 8.2 The Induced Character 253 8.3 Induction Workout 254 8.4 Universality 257 8.5 Universal Consequences 258 8.6 Reciprocity 260 8.7 Afterthoughts: Numbers 262 Exercises 263 Chapter 9 Commutant Duality 264 9.1 The Commutant 264 9.2 The Double Commutant 266 9.3 Commutant Decomposition of a Module 269 9.4 The Matrix Version 275 Exercises 279 Chapter 10 Character Duality 282 10.1 The Commutant for Sn on Vn 282 10.2 Schur–Weyl Duality 284 10.3 Character Duality, the High Road 285 10.4 Character Duality by Calculations 286 Exercises 293 Chapter 11 Representations of U(N) 295 11.1 The Haar Integral 296 11.2 The Weyl Integration Formula 297 11.3 Character Orthogonality 298 11.4 Weights 299 11.5 Characters of U(N) 300 11.6 Weyl Dimension Formula 304 11.7 From Weights to Representations 305 11.8 Characters of Sn from Characters of U(N) 308 Exercises 313 Chapter 12 Postscript: Algebra 315 12.1 Groups and Less 315 12.2 Rings and More 320 12.3 Fields 328 12.4 Modules over Rings 329 12.5 Free Modules and Bases 333 12.6 Power Series and Polynomials 337 12.7 Algebraic Integers 343 12.8 Linear Algebra 344 12.9 Tensor Products 349 12.10 Extension of Base Ring 352 12.11 Determinants and Traces of Matrices 353 12.12 Exterior Powers 354 12.13 Eigenvalues and Eigenvectors 359 12.14 Topology, Integration, and Hilbert Spaces 360 Bibliography 366 Index 373 This Graduate Textbook Presents The Basics Of Representation Theory For Finite Groups From The Point Of View Of Semisimple Algebras And Modules Over Them. The Presentation Interweaves Insights From Specific Examples With Development Of General And Powerful Tools Based On The Notion Of Semisimplicity. The Elegant Ideas Of Commutant Duality Are Introduced, Along With An Introduction To Representations Of Unitary Groups. the Text Progresses Systematically And The Presentation Is Friendly And Inviting. Central Concepts Are Revisited And Explored from Multiple Viewpoints. Exercises At The End Of The Chapter Help Reinforce The Material. Representing Finite Groups: A Semisimple Introduction Would Serve As A Textbook For Graduate And Some Advanced Undergraduate Courses In Mathematics. Prerequisites Include Acquaintance With Elementary Group Theory And Some Familiarity With Rings And Modules. A Final Chapter Presents A Self-contained Account Of Notions And Results In Algebra That Are Used. Researchers In Mathematics And Mathematical Physics Will Also Find This Book Useful. A Separate Solutions Manual Is Available For Instructors. 1. Concepts And Constructs -- 2. Basic Examples -- 3. The Group Algebra -- 4. More Group Algebra -- 5. Simply Semisimple -- 6. Representations Of S[subscript N] -- 7. Characters -- 8. Induced Representations -- 9. Commutant Duality -- 10. Character Duality -- 11. Representations Of U(n) -- 12. Postcript: Algebra. Ambar N. Sengupta. Includes Bibliographical References (p. 353-359) And Index. Front Matter....Pages i-xv Concepts and Constructs....Pages 1-38 Basic Examples....Pages 39-58 The Group Algebra....Pages 59-81 More Group Algebra....Pages 83-123 Simply Semisimple....Pages 125-156 Representations of S n ....Pages 157-188 Characters....Pages 189-234 Induced Representations....Pages 235-248 Commutant Duality....Pages 249-266 Character Duality....Pages 267-279 Representations of U ( N )....Pages 281-300 Postscript: Algebra....Pages 301-351 Back Matter....Pages 353-371 Annotation This text covers the basics of representation theory for finite groups from the viewpoint of semisimple algebras and modules over them. The presentation interweaves specific examples with the development of powerful tools based on the notion of semisimplicity