Representation theory of the symmetric groups

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 15
1.1.1 Representations......Page 19
1.1.2 Examples......Page 20
1.1.3 Intertwining operators......Page 22
1.1.4 Direct sums and complete reducibility......Page 23
1.1.5 The adjoint representation......Page 24
1.1.6 Matrix coefficients......Page 25
1.1.7 Tensor products......Page 26
1.1.8 Cyclic and invariant vectors......Page 28
1.2.1 Schur's lemma......Page 29
1.2.2 Multiplicities and isotypic components......Page 30
1.2.3 Finite dimensional algebras......Page 32
1.2.4 The structure of the commutant......Page 34
1.2.5 Another description of HomG(W, V)......Page 36
1.3.1 The trace......Page 37
1.3.2 Central functions and characters......Page 38
1.3.3 Central projection formulas......Page 40
1.4.1 Wielandt's lemma......Page 45
1.4.3 Frobenius reciprocity for a permutation representation......Page 48
1.4.4 The structure of the commutant ofa permutation representation......Page 53
1.5.1 L(G) and the convolution......Page 55
1.5.2 The Fourier transform......Page 60
1.5.3 Algebras of bi-K-invariant functions......Page 64
1.6.1 Definitions and examples......Page 69
1.6.2 First properties of induced representations......Page 71
1.6.3 Frobenius reciprocity......Page 73
1.6.4 Mackey's lemma and the intertwining number theorem......Page 75
2.1.1 Conjugacy invariant functions......Page 77
2.1.2 Multiplicity-free subgroups......Page 82
2.1.3 Greenhalgebras......Page 83
2.2.1 Branching graphs and Gelfand-Tsetlin bases......Page 87
2.2.2 Gelfand-Tsetlin algebras......Page 89
2.2.3 Gelfand-Tsetlin bases for permutation representations......Page 93
3.1.1 Partitions and conjugacy classes in Sn......Page 97
3.1.3 Young tableaux......Page 99
3.1.4 Coxeter generators......Page 101
3.1.5 The content of a tableau......Page 103
3.1.6 The Young poset......Page 107
3.2 The Young–Jucys–Murphy elements and a Gelfand-Tsetlin basis for Sn......Page 109
3.2.2 Marked permutations......Page 110
3.2.3 Olshanskii's theorem......Page 113
3.2.4 A characterization of the YJM elements......Page 116
3.3.1 The weight of a Young basis vector......Page 118
3.3.2 The spectrum of the YJM elements......Page 120
3.3.3 Spec(n) = Cont(n)......Page 122
3.4.1 Young's seminormal form......Page 128
3.4.2 Young's orthogonal form......Page 130
3.4.3 The Murnaghan-Nakayama rule for a cycle......Page 134
3.4.4 The Young seminormal units......Page 136
3.5.1 Skew shapes......Page 139
3.5.2 Skew representations of the symmetric group......Page 141
3.5.3 Basic properties of the skew representations and Pieri's rule......Page 144
3.5.4 Skew hooks......Page 148
3.5.5 The Murnaghan-Nakayama rule......Page 150
3.6.1 The dominance and the lexicographic orders for partitions......Page 153
3.6.2 The Young modules......Page 156
3.6.3 The Frobenius-Young correspondence......Page 158
3.6.4 Radon transforms between Young's modules......Page 162
3.7.1 Semistandard Young tableaux......Page 163
3.7.2 The reduced Young poset......Page 166
3.7.3 The Young rule......Page 168
3.7.4 A Greenhalgebra with the symmetric group......Page 171
4.1.1 More notation and results on partitions......Page 174
4.1.2 Monomial symmetric polynomials......Page 175
4.1.3 Elementary, complete and power sumssymmetric polynomials......Page 177
4.1.4 The fundamental theorem on symmetric polynomials......Page 183
4.1.5 An involutive map......Page 185
4.1.6 Antisymmetric polynomials......Page 186
4.1.7 The algebra of symmetric functions......Page 188
4.2.1 On the characters of the Young modules......Page 189
4.2.2 Cauchy's formula......Page 191
4.2.3 Frobenius character formula......Page 192
4.2.4 Applications of Frobenius character formula......Page 197
4.3.1 Definition of Schur polynomials......Page 203
4.3.2 A scalar product......Page 206
4.3.3 The characteristic map......Page 207
4.3.4 Determinantal identities......Page 211
4.4.1 Minimal decompositions of permutations asproducts of transpositions......Page 217
4.4.2 The Theorem of Jucys and Murphy......Page 222
4.4.3 Bernoulli and Stirling numbers......Page 226
4.4.4 Garsia's expression for chilambda......Page 231
5.1.1 Ordinary binomial coefficients: basic identities......Page 239
5.1.2 Binomial coefficients: some technical results......Page 242
5.1.3 Lassalle's coefficients......Page 246
5.1.4 Binomial coefficients associated with partitions......Page 251
5.1.5 Lassalle's symmetric function......Page 253
5.2.1 The Frobenius function......Page 256
5.2.2 Lagrange interpolation formula......Page 260
5.2.3 The Taylor series at infinity for the Frobenius quotient......Page 263
5.2.4 Some explicit formulas for the coefficients…......Page 268
5.3.1 Conjugacy classes with one nontrivial cycle......Page 270
5.3.2 Conjugacy classes with two nontrivial cycles......Page 272
5.3.3 The explicit formula for an arbitrary conjugacy class......Page 276
5.4 Central characters and class symmetric functions......Page 281
5.4.1 Central characters......Page 282
5.4.2 Class symmetric functions......Page 285
5.4.3 Kerov-Vershik asymptotics......Page 289
6 Radon transforms, Specht modules and the Littlewood–Richardson rule......Page 291
6.1.1 Words and lattice permutations......Page 292
6.1.2 Pairs of partitions......Page 295
6.1.3 James' combinatorial theorem......Page 299
6.1.4 Littlewood-Richardson tableaux......Page 302
6.1.5 The Littlewood-Richardson rule......Page 308
6.2.1 Generalized Specht modules......Page 311
6.2.2 A family of Radon transforms......Page 316
6.2.3 Decomposition theorems......Page 321
6.2.4 The Gelfand-Tsetlin bases for Ma revisited......Page 325
7.1.1 Finite dimensional *-algebras......Page 332
7.1.2 Burnside's theorem......Page 334
7.2.1 Schur's lemma for a linear algebra......Page 336
7.2.2 The commutant of a *-algebra......Page 338
7.3.1 Tensor product of algebras......Page 341
7.3.2 The double commutant theorem......Page 343
7.3.3 Structure of finite dimensional *-algebras......Page 345
7.3.4 Matrix units and central elements......Page 349
7.4.1 Representation theory of End(V)......Page 350
7.4.2 Representation theory of finite dimensional *-algebras......Page 352
7.4.4 Complete reducibility of finite dimensional *-algebras......Page 354
7.4.5 The regular representation of a *-algebra......Page 356
7.4.6 Representation theory of finite groups revisited......Page 357
7.5.1 Subalgebras and Bratteli diagrams......Page 359
7.5.2 The centralizer of a subalgebra......Page 361
7.5.3 A reciprocity law for restriction......Page 363
7.5.4 A reciprocity law for induction......Page 365
7.5.5 Iterated tensor product of permutation representations......Page 369
8.1 Symmetric and antisymmetric tensors......Page 375
8.1.1 Iterated tensor product......Page 376
8.1.2 The action of Sk on…......Page 378
8.1.3 Symmetric tensors......Page 379
8.1.4 Antisymmetric tensors......Page 383
8.2.1 The general linear group GL(n, C)......Page 386
8.2.2 Duality between GL(n,C) and Sk......Page 392
8.2.3 Clebsch-Gordan decomposition and branching formulas......Page 396
8.3 The partition algebra......Page 402
8.3.1 The partition monoid......Page 403
8.3.2 The partition algebra......Page 409
8.3.3 Schur-Weyl duality for the partition algebra......Page 411
References......Page 420
Index......Page 427