Preface page ix
PART I: LINEAR REPRESENTATIONS 1
1 Notation and generalities 3
2 Symmetric groups I 7
2.1 Gelfand–Zetlin bases 7
2.2 Description of weights 12
2.3 Formulas of Young and Murnaghan–Nakayama 17
3 Degenerate affine Hecke algebra 24
3.1 The algebras 25
3.2 Basis Theorem 26
3.3 The center of n 27
3.4 Parabolic subalgebras 28
3.5 Mackey Theorem 29
3.6 Some (anti) automorphisms 31
3.7 Duality 31
3.8 Intertwining elements 34
4 First results on n-modules 35
4.1 Formal characters 36
4.2 Central characters 37
4.3 Kato’s Theorem 38
4.4 Covering modules 40
5 Crystal operators 43
5.1 Multiplicity-free socles 44
5.2 Operators e˜a and f˜a 47
v
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0521837030 - Linear and Projective Representations of Symmetric Groups
Alexander Kleshchev
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vi Contents
5.3 Independence of irreducible characters 49
5.4 Labels for irreducibles 51
5.5 Alternative descriptions of a 51
6 Character calculations 54
6.1 Some irreducible induced modules 54
6.2 Calculations for small rank 57
6.3 Higher crystal operators 60
7 Integral representations and cyclotomic Hecke algebras 64
7.1 Integral representations 65
7.2 Some Lie theoretic notation 66
7.3 Degenerate cyclotomic Hecke algebras 68
7.4 The ∗-operation 69
7.5 Basis Theorem for cyclotomic Hecke algebras 70
7.6 Cyclotomic Mackey Theorem 73
7.7 Duality for cyclotomic algebras 74
7.8 Presentation for degenerate cyclotomic Hecke algebras 80
8 Functors e
i and f
i 82
8.1 New notation for blocks 83
8.2 Definitions 83
8.3 Divided powers 87
8.4 Functions
i 90
8.5 Alternative descriptions of
i 92
8.6 More on endomorphism algebras 99
9 Construction of U+ and irreducible modules 103
9.1 Grothendieck groups 104
9.2 Hopf algebra structure 106
9.3 Contravariant form 109
9.4 Chevalley relations 112
9.5 Identification of K∗, K∗, and K 115
9.6 Blocks 117
10 Identification of the crystal 120
10.1 Final properties of B 120
10.2 Crystals 123
10.3 Identification of B and B 126
11 Symmetric groups II 131
11.1 Description of the crystal graph 131
11.2 Main results on Sn 136
© Cambridge University Press www.cambridge.org
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0521837030 - Linear and Projective Representations of Symmetric Groups
Alexander Kleshchev
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Contents vii
PART II: PROJECTIVE REPRESENTATIONS 149
12 Generalities on superalgebra 151
12.1 Superalgebras and supermodules 151
12.2 Schur’s Lemma and Wedderburn’s Theorem 157
13 Sergeev superalgebras 165
13.1 Twisted group algebras 166
13.2 Sergeev superalgebras 168
14 Affine Sergeev superalgebras 174
14.1 The superalgebras 174
14.2 Basis Theorem for n 175
14.3 The center of n 176
14.4 Parabolic subalgebras of n 177
14.5 Mackey Theorem for n 177
14.6 Some (anti) automorphisms of n 178
14.7 Duality for n-supermodules 179
14.8 Intertwining elements for n 179
15 Integral representations and cyclotomic
Sergeev algebras 181
15.1 Integral representations of n 181
15.2 Some Lie theoretic notation 183
15.3 Cyclotomic Sergeev superalgebras 184
15.4 Basis Theorem for cyclotomic Sergeev
superalgebras 185
15.5 Cyclotomic Mackey Theorem 187
15.6 Duality for cyclotomic superalgebras 188
16 First results on n-modules 191
16.1 Formal characters of n-modules 191
16.2 Central characters and blocks 193
16.3 Kato’s Theorem for n 194
16.4 Covering modules for n 197
17 Crystal operators for n 200
17.1 Multiplicity-free socles 200
17.2 Operators e˜i and f˜i 203
17.3 Independence of irreducible characters 204
17.4 Labels for irreducibles 205
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Cambridge University Press
0521837030 - Linear and Projective Representations of Symmetric Groups
Alexander Kleshchev
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viii Contents
18 Character calculations for n 206
18.1 Some irreducible induced supermodules 206
18.2 Calculations for small rank 208
18.3 Higher crystal operators 216
19 Operators e
i and f
i 219
19.1 i-induction and i-restriction 219
19.2 Operators e
i and f
i 221
19.3 Divided powers 225
19.4 Alternative descriptions of i 228
19.5 The ∗-operation 229
19.6 Functions
i 229
19.7 Alternative descriptions of
i 230
20 Construction of U+ and irreducible modules 238
20.1 Grothendieck groups revisited 238
20.2 Hopf algebra structure 239
20.3 Shapovalov form 241
20.4 Chevalley relations 244
20.5 Identification of K∗, K∗ and K 246
20.6 Blocks of cyclotomic Sergeev superalgebras 247
21 Identification of the crystal 248
22 Double covers 250
22.1 Description of the crystal graph 250
22.2 Representations of Sergeev superalgebras 255
22.3 Spin representations of Sn 259
References 270
Index 275