Linear and Projective Representations of Symmetric Groups

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

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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

© Cambridge University Press www.cambridge.org

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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