Representing Finite Groups: A Semisimple Introduction (web version, 18 October 2011)

Preface......Page 3
Concepts and Constructs......Page 11
Representations of Groups......Page 12
Direct Sums and Tensor Products......Page 14
Change of Field......Page 15
Invariant Subspaces and Quotients......Page 16
Dual Representations......Page 17
Irreducible Representations......Page 20
Schur's Lemma......Page 22
The Frobenius-Schur Indicator......Page 24
Character of a Representation......Page 26
Diagonalizability......Page 30
Unitarity......Page 32
Rival Reads......Page 34
Afterthoughts: Lattices......Page 35
Exercises......Page 38
Reckoning......Page 46
Basic Examples......Page 49
Cyclic Groups......Page 50
Dihedral Groups......Page 53
The Symmetric Group S4......Page 58
Quaternionic Units......Page 62
Afterthoughts: Geometric Groups......Page 64
Exercises......Page 66
The Group Algebra......Page 69
Definition of the Group Algebra......Page 70
Representations of G and F[G]......Page 71
The Center......Page 73
Deconstructing F[S3]......Page 74
When F[G] is Semisimple......Page 83
Afterthoughts: Invariants......Page 87
Exercises......Page 89
More Group Algebra......Page 93
Looking Ahead......Page 94
Submodules and Idempotents......Page 96
Deconstructing F[G], the Module......Page 99
Deconstructing F[G], the Algebra......Page 101
As Simple as Matrix Algebras......Page 107
Putting F[G] back together......Page 111
The Mother of All Representations......Page 117
The Center......Page 119
Representing Abelian Groups......Page 121
Indecomposable Idempotents......Page 122
Beyond Our Borders......Page 124
Exercises......Page 125
Simply Semisimple......Page 135
Schur's Lemma......Page 136
Semisimple Modules......Page 137
Deconstructing Semisimple Modules......Page 141
Simple Modules for Semisimple Rings......Page 144
Deconstructing Semisimple Rings......Page 146
Simply Simple......Page 150
Commutants and Double Commutants......Page 151
Artin-Wedderburn Structure......Page 154
A Module as Sum of its Parts......Page 155
Afterthoughts: Clifford Algebras......Page 157
Exercises......Page 161
Permutations and Partitions......Page 167
Complements and Young Tableaux......Page 171
Symmetries of Partitions......Page 175
Conjugacy Classes to Young Tableaux......Page 178
Young Tableaux to Young Symmetrizers......Page 179
Youngtabs to Irreducible Representations......Page 180
Youngtab Apps......Page 184
Orthogonality......Page 189
Deconstructing F[Sn]......Page 190
Integrality......Page 193
Afterthoughts: Reflections......Page 194
Exercises......Page 198
Characters......Page 199
The Regular Character......Page 200
Character Orthogonality......Page 204
Character Expansions......Page 213
Comparing Z-Bases......Page 216
Character Arithmetic......Page 218
Computing Characters......Page 221
Return of the Group Determinant......Page 224
Orthogonality for Matrix Elements......Page 227
Solving Equations in Groups......Page 229
Afterthoughts: Connections......Page 238
Exercises......Page 239
Constructions......Page 245
The Induced Character......Page 248
Induction Workout......Page 249
Universality......Page 252
Universal Consequences......Page 253
Reciprocity......Page 255
Afterthoughts: Numbers......Page 257
Exercises......Page 258
The Commutant......Page 259
The Double Commutant......Page 261
Commutant Decomposition of a Module......Page 264
The Matrix Version......Page 270
Exercises......Page 274
The Commutant for Sn on Vn......Page 277
Schur-Weyl Duality......Page 279
Character Duality, the High Road......Page 280
Character Duality by Calculations......Page 281
Exercises......Page 288
Representations of U(N)......Page 291
The Haar Integral......Page 292
The Weyl Integration Formula......Page 293
Character Orthogonality......Page 294
Weights......Page 295
Characters of U(N)......Page 296
Weyl Dimension Formula......Page 300
From Weights to Representations......Page 301
Characters of Sn from Characters of U(N)......Page 304
Exercises......Page 308
Groups and Less......Page 311
Rings and More......Page 316
Fields......Page 324
Modules over Rings......Page 325
Free Modules and Bases......Page 329
Power Series and Polynomials......Page 333
Algebraic Integers......Page 339
Linear Algebra......Page 340
Tensor Products......Page 345
Extension of Base Ring......Page 348
Determinants and Traces of Matrices......Page 349
Exterior Powers......Page 350
Topology, Integration and Hilbert Spaces......Page 355
Selected Solutions......Page 361
Bibliography......Page 378