Holdings: Modular representations of finite groups of lie type

Modular representations of finite groups of lie type:

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Bibliographic Details
Main Author:	Humphreys, James E. 1939-2020 (Author)
Format:	Book
Language:	English
Published:	Cambridge Cambridge Univ. Pr. 2006
Edition:	1. publ.
Series:	London Mathematical Society lecture note series 326
Subjects:	Endliche Lie-Gruppe - Modulare Darstellung Finite simple groups Lie groups Modular representations of groups Endliche Gruppe Lie-Gruppe
Online Access:	Inhaltsverzeichnis
Physical Description:	XV, 233 S. graph. Darst.
ISBN:	0521674549 9780521674546

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adam_text	LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES 326 MODULAR REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE JAMES E. HUMPHREYS UNIVERSITY OF MASSACHUSETTS, AMHERST CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE PAGE XIII 1 FINITE GROUPS OF LIE TYPE 1 1.1 ALGEBRAIC GROUPS OVER FINITE FIELDS 1 1.2 CLASSIFICATION OVER FINITE FIELDS 2 1.3 FROBENIUS MAPS 3 1.4 LANG MAPS 4 1.5 CHEVALLEY GROUPS AND TWISTED GROUPS 5 1.6 EXAMPLE: SL(3, Q ) AND SU(3, Q ) 5 1.7 GROUPS WITH A BN-PAIR 7 1.8 NOTATIONAL CONVENTIONS 8 2 SIMPLE MODULES 9 2.1 REPRESENTATIONS AND FORMAL CHARACTERS 9 2.2 SIMPLE MODULES FOR ALGEBRAIC GROUPS 10 2.3 CONSTRUCTION OF MODULES T 11 2.4 CONTRAVARIANT FORMS 12 2.5 REPRESENTATIONS OF FROBENIUS KERNELS 13 2.6 INVARIANTS IN THE FUNCTION ALGEBRA 14 2.7 STEINBERG S TENSOR PRODUCT THEOREM 15 2.8 EXAMPLE: SL(2, K) . 16 2.9 BRAUER THEORY 16 2.10 COUNTING SEMISIMPLE CLASSES 16 2.11 RESTRICTION TO FINITE SUBGROUPS 17 2.12 PROOF OF IRREDUCIBILITY 17 2.13 PROOF OF DISTINCTNESS: CHEVALLEY GROUPS 18 2.14 PROOF OF DISTINCTNESS: TWISTED GROUPS 19 2.15 ACTION OF A SYLOW P-SUBGROUP 20 VI CONTENTS 3 WEYL MODULES AND LUSZTIG S CONJECTURE 21 3.1 WEYL MODULES 21 3.2 RESTRICTED HIGHEST WEIGHTS 22 3.3 COHOMOLOGY OF LINE BUNDLES 23 3.4 THE AFFINE WEYL GROUP 24 3.5 ALCOVES 25 3.6 LINKAGE AND TRANSLATION 26 3.7 STEINBERG MODULES 26 3.8 CONTRAVARIANT FORM ON A WEYL MODULE 27 3.9 JANTZEN FILTRATION AND SUM FORMULA 28 3.10 GENERIC BEHAVIOR OF WEYL MODULES 29 3.11 LUSZTIG S CONJECTURE 30 3.12 EVIDENCE FOR THE CONJECTURE 31 4 COMPUTATION OF WEIGHT MULTIPLICITIES 33 4.1 WEIGHT SPACES IN VERMA MODULES 33 4.2 WEIGHT SPACES IN CHARACTERISTIC P 34 4.3 AN EASY EXAMPLE: SL(2, K) 35 4.4 COMPUTATIONAL ALGORITHMS 35 4.5 FUNDAMENTAL MODULES FOR SYMPLECTIC GROUPS 37 4.6 SMALL WEIGHTS AND SMALL CHARACTERISTICS 38 4.7 SMALL REPRESENTATIONS 38 5 OTHER ASPECTS OF SIMPLE MODULES 41 5.1 RESTRICTION OF FROBENIUS MAPS 41 5.2 SPLITTING FIELDS 42 5.3 SPECIAL ISOGENIES 43 5.4 STEINBERG S REFINED FACTORIZATION 44 5.5 BRAUER CHARACTERS AND GROTHENDIECK RINGS 45 5.6 FORMAL CHARACTERS AND BRAUER CHARACTERS 46 5.7 REWRITING FORMAL SUMS 48 5.8 RESTRICTING HIGHEST WEIGHT MODULES TO FINITE SUBGROUPS 48 5.9 RESTRICTION OF WEYL MODULES 49 5.10 RESTRICTION OF SIMPLE MODULES TO LEVI SUBGROUPS 50 5.11 RESTRICTION TO ELEMENTARY ABELIAN P-SUBGROUPS 51 6 TENSOR PRODUCTS 53 6.1 TENSOR PRODUCTS OF SIMPLE MODULES 53 6.2 MULTIPLICITIES 54 CONTENTS VII 6.3 SIMPLE TENSOR PRODUCTS 55 6.4 SEMISIMPLE TENSOR PRODUCTS 56 6.5 DIMENSIONS DIVISIBLE BY P 56 6.6 FORMAL CHARACTERS AND MULTIPLICITIES * - - 57 6.7 TWISTING BY THE FROBENIUS 57 6.8 REWRITING MULTIPLICITIES 58 6.9 MULTIPLICITY OF THE STEINBERG MODULE 59 7 IW-PAIRS AND INDUCED MODULES 61 7.1 WEIGHTS FOR GROUPS WITH A SPLIT BA^-PAIR 61 7.2 PRINCIPAL SERIES AND INTERTWINING OPERATORS 63 7.3 EXAMPLES 64 7.4 SUMMANDS OF PRINCIPAL SERIES MODULES 64 7.5 HOMOLOGY REPRESENTATIONS 65 7.6 COMPARISON WITH PRINCIPAL SERIES 66 8 BLOCKS 67 8.1 BLOCKS OF A GROUP ALGEBRA 67 8.2 THE DEFECT OF A BLOCK 68 8.3 GROUPS OF LIE TYPE 68 8.4 DEFECT GROUPS 69 8.5 DEFECT GROUPS FOR GROUPS OF LIE TYPE 70 8.6 STEPS IN THE PROOF 70 8.7 THE BN-VAIR SETTING 72 8.8 VERTICES OF SIMPLE MODULES 72 8.9 REPRESENTATION TYPE OF A BLOCK 73 9 PROJECTIVE MODULES 75 9.1 PROJECTIVE MODULES FOR FINITE GROUPS 75 9.2 GROUPS OF LIE TYPE 76 9.3 THE STEINBERG MODULE 77 9.4 TENSORING WITH THE STEINBERG MODULE 78 9.5 BRAUER CHARACTERS AND ORTHOGONALITY RELATIONS 79 9.6 BRAUER CHARACTERS OF PIMS 79 9.7 A LOWER BOUND FOR DIMENSIONS OF PIMS 80 9.8 PROJECTIVE MODULES FOR SL(2, P) 81 9.9 BRAUER TREES FOR SL(2, P) 83 9.10 INDECOMPOSABLE MODULES FOR SL(2, P) 83 9.11 DIMENSIONS OF PIMS IN LOW RANKS 84 VIII CONTENTS 10 COMPARISON WITH FROBENIUS KERNELS 87 10.1 INJECTIVE MODULES FOR FROBENIUS KERNELS _ 87 10.2 TORUS ACTION ON G R -MODULES 89 10.3 FORMAL CHARACTER OF Q R (X) 90 10.4 LIFTING PIMS TO THE ALGEBRAIC GROUP 91 10.5 SOME CONSEQUENCES 92 10.6 TENSORING WITH THE STEINBERG MODULE 93 10.7 SMALL PIMS 94 10.8 THE CATEGORY OF G R T-MODULES 94 10.9 REWRITING MULTIPLICITIES 95 10.10 BRAUER CHARACTERS 96 10.11 STATEMENT OF THE MAIN THEOREM 96 10.12 PROOF OF THE MAIN THEOREM 97 10.13 LETTING R GROW 97 10.14 SOME COMPARISONS IN LOW RANKS 99 11 CARTAN INVARIANTS 101 11.1 CARTAN INVARIANTS FOR FINITE GROUPS 101 11.2 BRAUER CHARACTERS AND CARTAN INVARIANTS 102 11.3 DECOMPOSITION NUMBERS 102 11.4 GROUPS OF LIE TYPE 103 11.5 EXAMPLE: SL(2, P) 103 11.6 EXAMPLE: SL(2, Q) 104 11.7 USING STANDARD CHARACTER DATA TO COMPUTE CARTAN INVARIANTS 105 11.8 CONDITIONS FOR GENERICITY 105 11.9 GENERIC CARTAN INVARIANTS 106 11.10 GROWTH OF CARTAN INVARIANTS 107 11.11 THE FIRST CARTAN INVARIANT 108 11.12 SPECIAL CASES 108 11.13 COMPUTATIONS OF CARTAN MATRICES 109 11.14 EXAMPLE: SL(3, 3) 110 11.15 EXAMPLE: SP(4, 3) AND RELATED GROUPS 111 11.16 EXAMPLE: SL(5, 2) 111 11.17 CONJECTURES ON BLOCK INVARIANTS 112 12 EXTENSIONS OF SIMPLE MODULES 115 12.1 THE EXTENSION PROBLEM 115 12.2 EXAMPLE: SL(2, P) 116 12.3 THE OPTIMAL SITUATION 117 12.4 EXTENSIONS FOR ALGEBRAIC GROUPS 118 CONTENTS IX 12.5 INJ ECTIVITY THEOREM 12.6 DIMENSIONS OF EXT GROUPS 12.7 THE GENERIC CASE 12.8 TRUNCATED MODULE CATEGORIES AND COHOMOLOGY 12.9 COMPARISON THEOREMS 12.10 SELF-EXTENSIONS 12.11 SPECIAL CASES 12.12 SEMISIMPLICITY CRITERIA 13 LOEWY SERIES 13.1 LOEWY SERIES 13.2 LOEWY SERIES FOR FINITE GROUPS 13.3 EXAMPLE: SL(2, P) 13.4 MINIMAL PROJECTIVE RESOLUTIONS 13.5 EXAMPLE: SL(2, Q) 13.6 THE CATEGORY O 13.7 ANALOGIES IN CHARACTERISTIC P 13.8 FROBENIUS KERNELS AND ALGEBRAIC GROUPS 13.9 EXAMPLE: SL(3, K) 13.10 PRINCIPAL SERIES MODULES 13.11 LOEWY SERIES FOR CHEVALLEY GROUPS 13.12 EXAMPLE: SL(3, 2) 13.13 EXAMPLE: SL(3, 3) 13.14 EXAMPLE: SL(4, 2) 13.15 EXAMPLE: SO(5, 3) 14 COHOMOLOGY 14.1 COHOMOLOGY OF FINITE GROUPS 14.2 FINITE GROUPS OF LIE TYPE 14.3 COHOMOLOGY OF ALGEBRAIC GROUPS 14.4 TWISTING BY FROBENIUS MAPS 14.5 RATIONAL AND GENERIC COHOMOLOGY 14.6 DISCUSSION OF THE PROOF 14.7 EXAMPLE: SL(2, Q) 14.8 EXPLICIT COMPUTATIONS 14.9 RECENT DEVELOPMENTS 15 COMPLEXITY AND SUPPORT VARIETIES 15.1 COMPLEXITY OF A MODULE 15.2 SUDDORT VARIETIES 119 120 121 121 122 123 124 127 129 129 130 130 131 132 132 134 134 135 136 137 137 138 139 140 143 143 144 145 146 146 147 148 149 150 151 151 152 CONTENTS 15.3 SUPPORT VARIETIES FOR RESTRICTED LIE ALGEBRAS - 153 15.4 SUPPORT VARIETIES FOR GROUPS OF LIE TYPE 154 15.5 FURTHER REFINEMENTS 155 15.6 RESOLUTIONS AND PERIODICITY 156 15.7 PERIODIC MODULES FOR GROUPS OF LIE TYPE 156 16 ORDINARY AND MODULAR REPRESENTATIONS 159 16.1 THE DECOMPOSITION MATRIX 159 16.2 BRAUER CHARACTERS 160 16.3 BLOCKS 161 16.4 THE CARTAN-BRAUER TRIANGLE 161 16.5 GROUPS OF LIE TYPE 162 16.6 BLOCKS OF DEFECT ZERO: THE STEINBERG CHARACTER 162 16.7 CYCLIC BLOCKS AND BRAUER TREES 163 16.8 CHARACTERS OF SL(2,Q) 164 16.9 THE BRAUER TREE OF SL(2, P) 165 16.10 DECOMPOSITION NUMBERS OF SL(2, Q) 165 16.11 CHARACTER COMPUTATIONS FOR LIE FAMILIES 167 16.12 SOME EXPLICIT DECOMPOSITION MATRICES 168 16.13 SPECIAL CHARACTERS OF SP(2N, 16.14 IRREDUCIBILITY MODULO P 169 17 DELIGNE-LUSZTIG CHARACTERS 171 17.1 REDUCTIVE GROUPS AND FROBENIUS MAPS 171 17.2 F-STABLE MAXIMAL TORI. 172 17.3 DL CHARACTERS 173 17.4 BASIC PROPERTIES OF DL CHARACTERS 174 17.5 THE DECOMPOSITION PROBLEM 175 17.6 PIMS AND DL CHARACTERS 176 17.7 DL CHARACTERS AND WEYL CHARACTERS 176 17.8 GENERIC DECOMPOSITION PATTERNS 177 17.9 TWISTED GROUPS 178 17.10 UNIPOTENT CHARACTERS 178 17.11 SEMISIMPLE AND REGULAR CHARACTERS 179 17.12 GEOMETRIC CONJ UGACY CLASSES OF CARDINALITY 2 181 17.13 JANTZEN FILTRATION OF A PRINCIPAL SERIES MODULE 181 17.14 EXTREMAL COMPOSITION FACTORS 182 17.15 ANOTHER LOOK AT THE BRAUER TREE OF SL(2, P) 182 17.16 THE BRAUER COMPLEX 183 17.17 DUAL FORMULATION 184 CONTENTS XI 18 THE GROUPS G 2 (Q) 185 18.1 THE GROUPS 185 18.2 THE AFFINE WEYL GROUP 185 18.3 WEYL MODULES AND SIMPLE MODULES - 186 18.4 PROJECTIVE MODULES FOR P = 2,3,5 187 18.5 PROJECTIVE MODULES FOR P 7 191 18.6 CHARACTERS OF G 2 (Q) 191 18.7 REDUCTION MODULO P 194 18.8 BRAUER COMPLEX OF G 2 (5) 195 19 GENERAL AND SPECIAL LINEAR GROUPS 197 19.1 REPRESENTATIONS OF GL(N, K) AN D GL(N, Q) 197 19.2 ACTION ON SYMMETRIC POWERS 198 19.3 DICKSON INVARIANTS 198 19.4 COMPLEMENTS 199 19.5 MULTIPLICITY OF ST IN SYMMETRIC POWERS 200 19.6 OTHER SIMPLE MODULES 200 19.7 EXAMPLE: SL(2, P) 201 19.8 PERIODICITY FOR SL(2, Q) 201 19.9 KEY LEMMA 202 19.10 PROOF OF PERIODICITY THEOREM 202 19.11 BRAUER LIFTING 204 20 SUZUKI AND REE GROUPS 205 20.1 DESCRIPTION OF THE GROUPS 205 20.2 SIMPLE MODULES 206 20.3 PROJECTIVE MODULES AND BLOCKS 208 20.4 CARTAN INVARIANTS OF SUZUKI GROUPS 208 20.5 CARTAN INVARIANTS OF THE TITS GROUP 209 20.6 EXTENSIONS AND COHOMOLOGY 210 20.7 ORDINARY CHARACTERS 210 20.8 DECOMPOSITION NUMBERS OF SUZUKI GROUPS 211 BIBLIOGRAPHY 213 FREQUENTLY USED SYMBOLS 229 INDEX 231
adam_txt	LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES 326 MODULAR REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE JAMES E. HUMPHREYS UNIVERSITY OF MASSACHUSETTS, AMHERST CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE PAGE XIII 1 FINITE GROUPS OF LIE TYPE 1 1.1 ALGEBRAIC GROUPS OVER FINITE FIELDS 1 1.2 CLASSIFICATION OVER FINITE FIELDS 2 1.3 FROBENIUS MAPS 3 1.4 LANG MAPS 4 1.5 CHEVALLEY GROUPS AND TWISTED GROUPS 5 1.6 EXAMPLE: SL(3, Q ) AND SU(3, Q ) 5 1.7 GROUPS WITH A BN-PAIR 7 1.8 NOTATIONAL CONVENTIONS 8 2 SIMPLE MODULES 9 2.1 REPRESENTATIONS AND FORMAL CHARACTERS 9 2.2 SIMPLE MODULES FOR ALGEBRAIC GROUPS 10 2.3 CONSTRUCTION OF MODULES T 11 2.4 CONTRAVARIANT FORMS 12 2.5 REPRESENTATIONS OF FROBENIUS KERNELS 13 2.6 INVARIANTS IN THE FUNCTION ALGEBRA 14 2.7 STEINBERG'S TENSOR PRODUCT THEOREM 15 2.8 EXAMPLE: SL(2, K) . 16 2.9 BRAUER THEORY 16 2.10 COUNTING SEMISIMPLE CLASSES 16 2.11 RESTRICTION TO FINITE SUBGROUPS 17 2.12 PROOF OF IRREDUCIBILITY 17 2.13 PROOF OF DISTINCTNESS: CHEVALLEY GROUPS 18 2.14 PROOF OF DISTINCTNESS: TWISTED GROUPS 19 2.15 ACTION OF A SYLOW P-SUBGROUP 20 VI CONTENTS 3 WEYL MODULES AND LUSZTIG'S CONJECTURE 21 3.1 WEYL MODULES 21 3.2 RESTRICTED HIGHEST WEIGHTS 22 3.3 COHOMOLOGY OF LINE BUNDLES 23 3.4 THE AFFINE WEYL GROUP 24 3.5 ALCOVES 25 3.6 LINKAGE AND TRANSLATION 26 3.7 STEINBERG MODULES 26 3.8 CONTRAVARIANT FORM ON A WEYL MODULE 27 3.9 JANTZEN FILTRATION AND SUM FORMULA 28 3.10 GENERIC BEHAVIOR OF WEYL MODULES 29 3.11 LUSZTIG'S CONJECTURE 30 3.12 EVIDENCE FOR THE CONJECTURE 31 4 COMPUTATION OF WEIGHT MULTIPLICITIES 33 4.1 WEIGHT SPACES IN VERMA MODULES 33 4.2 WEIGHT SPACES IN CHARACTERISTIC P 34 4.3 AN EASY EXAMPLE: SL(2, K) 35 4.4 COMPUTATIONAL ALGORITHMS 35 4.5 FUNDAMENTAL MODULES FOR SYMPLECTIC GROUPS 37 4.6 SMALL WEIGHTS AND SMALL CHARACTERISTICS 38 4.7 SMALL REPRESENTATIONS 38 5 OTHER ASPECTS OF SIMPLE MODULES 41 5.1 RESTRICTION OF FROBENIUS MAPS 41 5.2 SPLITTING FIELDS 42 5.3 SPECIAL ISOGENIES 43 5.4 STEINBERG'S REFINED FACTORIZATION 44 5.5 BRAUER CHARACTERS AND GROTHENDIECK RINGS 45 5.6 FORMAL CHARACTERS AND BRAUER CHARACTERS 46 5.7 REWRITING FORMAL SUMS 48 5.8 RESTRICTING HIGHEST WEIGHT MODULES TO FINITE SUBGROUPS 48 5.9 RESTRICTION OF WEYL MODULES 49 5.10 RESTRICTION OF SIMPLE MODULES TO LEVI SUBGROUPS 50 5.11 RESTRICTION TO ELEMENTARY ABELIAN P-SUBGROUPS 51 6 TENSOR PRODUCTS 53 6.1 TENSOR PRODUCTS OF SIMPLE MODULES 53 6.2 MULTIPLICITIES 54 CONTENTS VII 6.3 SIMPLE TENSOR PRODUCTS 55 6.4 SEMISIMPLE TENSOR PRODUCTS 56 6.5 DIMENSIONS DIVISIBLE BY P 56 6.6 FORMAL CHARACTERS AND MULTIPLICITIES * - - 57 6.7 TWISTING BY THE FROBENIUS 57 6.8 REWRITING MULTIPLICITIES 58 6.9 MULTIPLICITY OF THE STEINBERG MODULE 59 7 IW-PAIRS AND INDUCED MODULES 61 7.1 WEIGHTS FOR GROUPS WITH A SPLIT BA^-PAIR 61 7.2 PRINCIPAL SERIES AND INTERTWINING OPERATORS 63 7.3 EXAMPLES 64 7.4 SUMMANDS OF PRINCIPAL SERIES MODULES 64 7.5 HOMOLOGY REPRESENTATIONS 65 7.6 COMPARISON WITH PRINCIPAL SERIES 66 8 BLOCKS 67 8.1 BLOCKS OF A GROUP ALGEBRA 67 8.2 THE DEFECT OF A BLOCK 68 8.3 GROUPS OF LIE TYPE 68 8.4 DEFECT GROUPS 69 8.5 DEFECT GROUPS FOR GROUPS OF LIE TYPE 70 8.6 STEPS IN THE PROOF 70 8.7 THE BN-VAIR SETTING 72 8.8 VERTICES OF SIMPLE MODULES 72 8.9 REPRESENTATION TYPE OF A BLOCK 73 9 PROJECTIVE MODULES 75 9.1 PROJECTIVE MODULES FOR FINITE GROUPS 75 9.2 GROUPS OF LIE TYPE 76 9.3 THE STEINBERG MODULE 77 9.4 TENSORING WITH THE STEINBERG MODULE 78 9.5 BRAUER CHARACTERS AND ORTHOGONALITY RELATIONS 79 9.6 BRAUER CHARACTERS OF PIMS 79 9.7 A LOWER BOUND FOR DIMENSIONS OF PIMS 80 9.8 PROJECTIVE MODULES FOR SL(2, P) 81 9.9 BRAUER TREES FOR SL(2, P) 83 9.10 INDECOMPOSABLE MODULES FOR SL(2, P) 83 9.11 DIMENSIONS OF PIMS IN LOW RANKS 84 VIII CONTENTS 10 COMPARISON WITH FROBENIUS KERNELS 87 10.1 INJECTIVE MODULES FOR FROBENIUS KERNELS _ 87 10.2 TORUS ACTION ON G R -MODULES 89 10.3 FORMAL CHARACTER OF Q R (X) 90 10.4 LIFTING PIMS TO THE ALGEBRAIC GROUP 91 10.5 SOME CONSEQUENCES 92 10.6 TENSORING WITH THE STEINBERG MODULE 93 10.7 SMALL PIMS 94 10.8 THE CATEGORY OF G R T-MODULES 94 10.9 REWRITING MULTIPLICITIES 95 10.10 BRAUER CHARACTERS 96 10.11 STATEMENT OF THE MAIN THEOREM 96 10.12 PROOF OF THE MAIN THEOREM 97 10.13 LETTING R GROW 97 10.14 SOME COMPARISONS IN LOW RANKS 99 11 CARTAN INVARIANTS 101 11.1 CARTAN INVARIANTS FOR FINITE GROUPS 101 11.2 BRAUER CHARACTERS AND CARTAN INVARIANTS 102 11.3 DECOMPOSITION NUMBERS 102 11.4 GROUPS OF LIE TYPE 103 11.5 EXAMPLE: SL(2, P) 103 11.6 EXAMPLE: SL(2, Q) 104 11.7 USING STANDARD CHARACTER DATA TO COMPUTE CARTAN INVARIANTS 105 11.8 CONDITIONS FOR GENERICITY 105 11.9 GENERIC CARTAN INVARIANTS 106 11.10 GROWTH OF CARTAN INVARIANTS 107 11.11 THE FIRST CARTAN INVARIANT 108 11.12 SPECIAL CASES 108 11.13 COMPUTATIONS OF CARTAN MATRICES 109 11.14 EXAMPLE: SL(3, 3) 110 11.15 EXAMPLE: SP(4, 3) AND RELATED GROUPS 111 11.16 EXAMPLE: SL(5, 2) 111 11.17 CONJECTURES ON BLOCK INVARIANTS 112 12 EXTENSIONS OF SIMPLE MODULES 115 12.1 THE EXTENSION PROBLEM 115 12.2 EXAMPLE: SL(2, P) 116 12.3 THE OPTIMAL SITUATION 117 12.4 EXTENSIONS FOR ALGEBRAIC GROUPS 118 CONTENTS IX 12.5 INJ ECTIVITY THEOREM 12.6 DIMENSIONS OF EXT GROUPS 12.7 THE GENERIC CASE 12.8 TRUNCATED MODULE CATEGORIES AND COHOMOLOGY 12.9 COMPARISON THEOREMS 12.10 SELF-EXTENSIONS 12.11 SPECIAL CASES 12.12 SEMISIMPLICITY CRITERIA 13 LOEWY SERIES 13.1 LOEWY SERIES 13.2 LOEWY SERIES FOR FINITE GROUPS 13.3 EXAMPLE: SL(2, P) 13.4 MINIMAL PROJECTIVE RESOLUTIONS 13.5 EXAMPLE: SL(2, Q) 13.6 THE CATEGORY O 13.7 ANALOGIES IN CHARACTERISTIC P 13.8 FROBENIUS KERNELS AND ALGEBRAIC GROUPS 13.9 EXAMPLE: SL(3, K) 13.10 PRINCIPAL SERIES MODULES 13.11 LOEWY SERIES FOR CHEVALLEY GROUPS 13.12 EXAMPLE: SL(3, 2) 13.13 EXAMPLE: SL(3, 3) 13.14 EXAMPLE: SL(4, 2) 13.15 EXAMPLE: SO(5, 3) 14 COHOMOLOGY 14.1 COHOMOLOGY OF FINITE GROUPS \ 14.2 FINITE GROUPS OF LIE TYPE 14.3 COHOMOLOGY OF ALGEBRAIC GROUPS 14.4 TWISTING BY FROBENIUS MAPS 14.5 RATIONAL AND GENERIC COHOMOLOGY 14.6 DISCUSSION OF THE PROOF 14.7 EXAMPLE: SL(2, Q) 14.8 EXPLICIT COMPUTATIONS 14.9 RECENT DEVELOPMENTS 15 COMPLEXITY AND SUPPORT VARIETIES 15.1 COMPLEXITY OF A MODULE 15.2 SUDDORT VARIETIES 119 120 121 121 122 123 124 127 129 129 130 130 131 132 132 134 134 135 136 137 137 138 139 140 143 143 144 145 146 146 147 148 149 150 151 151 152 CONTENTS 15.3 SUPPORT VARIETIES FOR RESTRICTED LIE ALGEBRAS - 153 15.4 SUPPORT VARIETIES FOR GROUPS OF LIE TYPE 154 15.5 FURTHER REFINEMENTS 155 15.6 RESOLUTIONS AND PERIODICITY 156 15.7 PERIODIC MODULES FOR GROUPS OF LIE TYPE 156 16 ORDINARY AND MODULAR REPRESENTATIONS 159 16.1 THE DECOMPOSITION MATRIX 159 16.2 BRAUER CHARACTERS 160 16.3 BLOCKS 161 16.4 THE CARTAN-BRAUER TRIANGLE 161 16.5 GROUPS OF LIE TYPE 162 16.6 BLOCKS OF DEFECT ZERO: THE STEINBERG CHARACTER 162 16.7 CYCLIC BLOCKS AND BRAUER TREES 163 16.8 CHARACTERS OF SL(2,Q) 164 16.9 THE BRAUER TREE OF SL(2, P) 165 16.10 DECOMPOSITION NUMBERS OF SL(2, Q) 165 16.11 CHARACTER COMPUTATIONS FOR LIE FAMILIES 167 16.12 SOME EXPLICIT DECOMPOSITION MATRICES 168 16.13 SPECIAL CHARACTERS OF SP(2N, 16.14 IRREDUCIBILITY MODULO P 169 17 DELIGNE-LUSZTIG CHARACTERS 171 17.1 REDUCTIVE GROUPS AND FROBENIUS MAPS 171 17.2 F-STABLE MAXIMAL TORI. 172 17.3 DL CHARACTERS 173 17.4 BASIC PROPERTIES OF DL CHARACTERS 174 17.5 THE DECOMPOSITION PROBLEM 175 17.6 PIMS AND DL CHARACTERS 176 17.7 DL CHARACTERS AND WEYL CHARACTERS 176 17.8 GENERIC DECOMPOSITION PATTERNS 177 17.9 TWISTED GROUPS 178 17.10 UNIPOTENT CHARACTERS 178 17.11 SEMISIMPLE AND REGULAR CHARACTERS 179 17.12 GEOMETRIC CONJ UGACY CLASSES OF CARDINALITY 2 181 17.13 JANTZEN FILTRATION OF A PRINCIPAL SERIES MODULE 181 17.14 EXTREMAL COMPOSITION FACTORS 182 17.15 ANOTHER LOOK AT THE BRAUER TREE OF SL(2, P) 182 17.16 THE BRAUER COMPLEX 183 17.17 DUAL FORMULATION 184 CONTENTS XI 18 THE GROUPS G 2 (Q) 185 18.1 THE GROUPS 185 18.2 THE AFFINE WEYL GROUP 185 18.3 WEYL MODULES AND SIMPLE MODULES - 186 18.4 PROJECTIVE MODULES FOR P = 2,3,5 187 18.5 PROJECTIVE MODULES FOR P 7 191 18.6 CHARACTERS OF G 2 (Q) 191 18.7 REDUCTION MODULO P 194 18.8 BRAUER COMPLEX OF G 2 (5) 195 19 GENERAL AND SPECIAL LINEAR GROUPS 197 19.1 REPRESENTATIONS OF GL(N, K) AN D GL(N, Q) 197 19.2 ACTION ON SYMMETRIC POWERS 198 19.3 DICKSON INVARIANTS 198 19.4 COMPLEMENTS 199 19.5 MULTIPLICITY OF ST IN SYMMETRIC POWERS 200 19.6 OTHER SIMPLE MODULES 200 19.7 EXAMPLE: SL(2, P) 201 19.8 PERIODICITY FOR SL(2, Q) 201 19.9 KEY LEMMA 202 19.10 PROOF OF PERIODICITY THEOREM 202 19.11 BRAUER LIFTING 204 20 SUZUKI AND REE GROUPS 205 20.1 DESCRIPTION OF THE GROUPS 205 20.2 SIMPLE MODULES 206 20.3 PROJECTIVE MODULES AND BLOCKS 208 20.4 CARTAN INVARIANTS OF SUZUKI GROUPS 208 20.5 CARTAN INVARIANTS OF THE TITS GROUP 209 20.6 EXTENSIONS AND COHOMOLOGY 210 20.7 ORDINARY CHARACTERS 210 20.8 DECOMPOSITION NUMBERS OF SUZUKI GROUPS 211 BIBLIOGRAPHY 213 FREQUENTLY USED SYMBOLS 229 INDEX 231
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dewey-ones	512 - Algebra
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discipline	Mathematik
discipline_str_mv	Mathematik
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format	Book
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id	DE-604.BV021519478
illustrated	Illustrated
index_date	2024-07-02T14:22:04Z
indexdate	2024-07-09T20:37:40Z
institution	BVB
isbn	0521674549 9780521674546
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-014735997
oclc_num	254776490
open_access_boolean
owner	DE-91G DE-BY-TUM DE-11 DE-703 DE-188
owner_facet	DE-91G DE-BY-TUM DE-11 DE-703 DE-188
physical	XV, 233 S. graph. Darst.
publishDate	2006
publishDateSearch	2006
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publisher	Cambridge Univ. Pr.
record_format	marc
series	London Mathematical Society lecture note series
series2	London Mathematical Society lecture note series
spelling	Humphreys, James E. 1939-2020 Verfasser (DE-588)108120848 aut Modular representations of finite groups of lie type James E. Humphreys 1. publ. Cambridge Cambridge Univ. Pr. 2006 XV, 233 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society lecture note series 326 Endliche Lie-Gruppe - Modulare Darstellung Finite simple groups Lie groups Modular representations of groups Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 s Lie-Gruppe (DE-588)4035695-4 s DE-604 London Mathematical Society lecture note series 326 (DE-604)BV000000130 326 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014735997&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Humphreys, James E. 1939-2020 Modular representations of finite groups of lie type London Mathematical Society lecture note series Endliche Lie-Gruppe - Modulare Darstellung Finite simple groups Lie groups Modular representations of groups Endliche Gruppe (DE-588)4014651-0 gnd Lie-Gruppe (DE-588)4035695-4 gnd
subject_GND	(DE-588)4014651-0 (DE-588)4035695-4
title	Modular representations of finite groups of lie type
title_auth	Modular representations of finite groups of lie type
title_exact_search	Modular representations of finite groups of lie type
title_exact_search_txtP	Modular representations of finite groups of lie type
title_full	Modular representations of finite groups of lie type James E. Humphreys
title_fullStr	Modular representations of finite groups of lie type James E. Humphreys
title_full_unstemmed	Modular representations of finite groups of lie type James E. Humphreys
title_short	Modular representations of finite groups of lie type
title_sort	modular representations of finite groups of lie type
topic	Endliche Lie-Gruppe - Modulare Darstellung Finite simple groups Lie groups Modular representations of groups Endliche Gruppe (DE-588)4014651-0 gnd Lie-Gruppe (DE-588)4035695-4 gnd
topic_facet	Endliche Lie-Gruppe - Modulare Darstellung Finite simple groups Lie groups Modular representations of groups Endliche Gruppe Lie-Gruppe
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014735997&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000000130
work_keys_str_mv	AT humphreysjamese modularrepresentationsoffinitegroupsoflietype

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