Linear and projective representations of symmetric groups:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge University Press
2005
|
| Edition: | 1. publ. |
| Series: | Cambridge tracts in mathematics
163 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 277 S. Ill. |
| ISBN: | 0521837030 |
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| 100 | 1 | |a Kleščev, Aleksandr S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Linear and projective representations of symmetric groups |c Alexander Kleshchev |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2005 | |
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| 650 | 4 | |a Modular representations of groups | |
| 650 | 4 | |a Hecke algebras | |
| 650 | 4 | |a Superalgebras | |
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Record in the Search Index
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| adam_text | Contents
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 Jin 21
3.4 Parabolic subalgebras 28
3.5 Mackey Theorem 29
s 3.6 Some (anti) automorphisms 31
ie 3.7 Duality 31
1 3.8 Intertwining elements 34
4 First results on ^ 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 eu and /„ 47
v
vi Contents
5.3 Independence of irreducible characters 49
5.4 Labels for irreducibles 51
5.5 Alternative descriptions of ea 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 ef and /* 82
8.1 New notation for blocks 83
8.2 Definitions 83
8.3 Divided powers 87
8.4 Functions (pf 90
8.5 Alternative descriptions of pf 92
8.6 More on endomorphism algebras 99
9 Construction of Ut 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{oo) K( ) and K( ) 115
9.6 Blocks 117
10 Identification of the crystal 120
10.1 Final properties of tf(oc) 120
10.2 Crystals 123
10.3 Identification of fl(oo) and fi(A) 126
11 Symmetric groups II 131
11.1 Description of the crystal graph 131
11.2 Main results on S,, 136
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 Xn 175
14.3 The center of X,, 176
14.4 Parabolic subalgebras of X,, 177
14.5 Mackey Theorem for X,, 177
14.6 Some (anti) automorphisms of Xn 178
14.7 Duality for Xn supermodules 179
14.8 Intertwining elements for Xn 179
15 Integral representations and cyclotomic
Sergeev algebras 181
15.1 Integral representations of Xn 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 X,, modules 191
16.1 Formal characters of .X,, modules 191
16.2 Central characters and blocks 193
16.3 Kato s Theorem for X,, 194
16.4 Covering modules for Xn 197
17 Crystal operators for Xn 200
17.1 Multiplicity free socles 200
17.2 Operators et and/, 203
17.3 Independence of irreducible characters 204
17.4 Labels for irreducibles 205
viii Contents
18 Character calculations for Xn 206
18.1 Some irreducible induced supermodules 206
18.2 Calculations for small rank 208
18.3 Higher crystal operators 216
19 Operators ef and /* 219
19.1 / induction and / restriction 219
19.2 Operators ?f and/,A 221
19.3 Divided powers 225
19.4 Alternative descriptions of e, 228
19.5 The * operation 229
19.6 Functions pf 229
19.7 Alternative descriptions of pf 230
20 Construction of l/J 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(oo)*, K(X)*, 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
|
| adam_txt |
Contents
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 Jin 21
3.4 Parabolic subalgebras 28
3.5 Mackey Theorem 29
s 3.6 Some (anti) automorphisms 31
ie 3.7 Duality 31
1 3.8 Intertwining elements 34
4 First results on ^ 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 eu and /„ 47
v
vi Contents
5.3 Independence of irreducible characters 49
5.4 Labels for irreducibles 51
5.5 Alternative descriptions of ea 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 ef and /* 82
8.1 New notation for blocks 83
8.2 Definitions 83
8.3 Divided powers 87
8.4 Functions (pf 90
8.5 Alternative descriptions of pf 92
8.6 More on endomorphism algebras 99
9 Construction of Ut 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{oo)\ K(\)\ and K(\) 115
9.6 Blocks 117
10 Identification of the crystal 120
10.1 Final properties of tf(oc) 120
10.2 Crystals 123
10.3 Identification of fl(oo) and fi(A) 126
11 Symmetric groups II 131
11.1 Description of the crystal graph 131
11.2 Main results on S,, 136
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 Xn 175
14.3 The center of X,, 176
14.4 Parabolic subalgebras of X,, 177
14.5 Mackey Theorem for X,, 177
14.6 Some (anti) automorphisms of Xn 178
14.7 Duality for Xn supermodules 179
14.8 Intertwining elements for Xn 179
15 Integral representations and cyclotomic
Sergeev algebras 181
15.1 Integral representations of Xn 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 X,, modules 191
16.1 Formal characters of .X,, modules 191
16.2 Central characters and blocks 193
16.3 Kato's Theorem for X,, 194
16.4 Covering modules for Xn 197
17 Crystal operators for Xn 200
17.1 Multiplicity free socles 200
17.2 Operators et and/, 203
17.3 Independence of irreducible characters 204
17.4 Labels for irreducibles 205
viii Contents
18 Character calculations for Xn 206
18.1 Some irreducible induced supermodules 206
18.2 Calculations for small rank 208
18.3 Higher crystal operators 216
19 Operators ef and /* 219
19.1 / induction and / restriction 219
19.2 Operators ?f and/,A 221
19.3 Divided powers 225
19.4 Alternative descriptions of e, 228
19.5 The * operation 229
19.6 Functions pf 229
19.7 Alternative descriptions of pf 230
20 Construction of l/J 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(oo)*, K(X)*, 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 |
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| spelling | Kleščev, Aleksandr S. Verfasser aut Linear and projective representations of symmetric groups Alexander Kleshchev 1. publ. Cambridge [u.a.] Cambridge University Press 2005 XIV, 277 S. Ill. txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 163 Representations of groups Symmetry groups Modular representations of groups Hecke algebras Superalgebras Linear algebraic groups Algebras, Linear Geometry, Projective Symmetrische Gruppe (DE-588)4184204-2 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Symmetrische Gruppe (DE-588)4184204-2 s Darstellungstheorie (DE-588)4148816-7 s DE-604 Cambridge tracts in mathematics 163 (DE-604)BV000000001 163 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014595280&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kleščev, Aleksandr S. Linear and projective representations of symmetric groups Cambridge tracts in mathematics Representations of groups Symmetry groups Modular representations of groups Hecke algebras Superalgebras Linear algebraic groups Algebras, Linear Geometry, Projective Symmetrische Gruppe (DE-588)4184204-2 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| subject_GND | (DE-588)4184204-2 (DE-588)4148816-7 |
| title | Linear and projective representations of symmetric groups |
| title_auth | Linear and projective representations of symmetric groups |
| title_exact_search | Linear and projective representations of symmetric groups |
| title_exact_search_txtP | Linear and projective representations of symmetric groups |
| title_full | Linear and projective representations of symmetric groups Alexander Kleshchev |
| title_fullStr | Linear and projective representations of symmetric groups Alexander Kleshchev |
| title_full_unstemmed | Linear and projective representations of symmetric groups Alexander Kleshchev |
| title_short | Linear and projective representations of symmetric groups |
| title_sort | linear and projective representations of symmetric groups |
| topic | Representations of groups Symmetry groups Modular representations of groups Hecke algebras Superalgebras Linear algebraic groups Algebras, Linear Geometry, Projective Symmetrische Gruppe (DE-588)4184204-2 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| topic_facet | Representations of groups Symmetry groups Modular representations of groups Hecke algebras Superalgebras Linear algebraic groups Algebras, Linear Geometry, Projective Symmetrische Gruppe Darstellungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014595280&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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