Reflection groups and Coxeter groups:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge u.a.
Cambridge Univ. Pr.
2000
|
| Edition: | 1. paperback ed. (with corr.), transferred to digital print. |
| Series: | Cambridge studies in advanced mathematics
29 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 185 - 201 |
| Physical Description: | XII, 204 S. graph. Darst. |
| ISBN: | 0521436133 9780521436137 |
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MARC
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| 245 | 1 | 0 | |a Reflection groups and Coxeter groups |c James E. Humphreys |
| 250 | |a 1. paperback ed. (with corr.), transferred to digital print. | ||
| 264 | 1 | |a Cambridge u.a. |b Cambridge Univ. Pr. |c 2000 | |
| 300 | |a XII, 204 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 29 | |
| 500 | |a Literaturverz. S. 185 - 201 | ||
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| 650 | 4 | |a Grupos finitos | |
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Record in the Search Index
| _version_ | 1804138613469872129 |
|---|---|
| adam_text | Contents
Contents
vii
Preface
xi
I Finite and
affine
reflection groups
1
1
Finite reflection
groupe
3
1.1
Reflections
........................... 3
1.2
Roots
............................. 6
1.3
Positive and simple systems
................. 7
1.4
Conjugacy of positive and simple systems
......... 10
1.5
Generation by simple reflections
............... 10
1.6
The length function
..................... 12
1.7
Deletion and Exchange Conditions
............. 13
1.8
Simple transitivity and the longest element
........ 15
1.9
Generators and relations
................... 16
1.10
Parabolic subgroups
..................... 18
1.11
Poincaré
polynomials
..................... 20
1.12
Fundamental domains
.................... 21
1.13
The lattice of parabolic subgroups
............. 24
1.14
Reflections in
W
....................... 24
1.15
The Coxeter complex
..................... 25
1.16
An alternating sum formula
................. 26
2
Classification of finite reflection
groupe
29
2.1
Isomorphisms
......................... 29
2.2
Irreducible components
................... 30
2.3
Coxeter graphs and associated bilinear forms
....... 31
2.4
Some positive definite graphs
................ 32
2.5
Some positive
semidefinite
graphs
.............. 33
2.6
Subgraphs
........................... 35
2.7
Classification of graphs of positive type
.......... 36
vu
Contents
2.8
Crystallographic groups
................... 38
2.9
Crystallographic
root systems and Weyl groups
...... 39
2.10
Construction of root systems
................ 41
2.11
Computing the order of
W
................. 43
2.12
Exceptional Weyl groups
................... 45
2.13
Groups of types H3 and H4
................. 46
Polynomial invariants of finite reflection groups
49
3.1
Polynomial invariants of a finite group
........... 49
3.2
Finite generation
....................... 50
3.3
A divisibility criterion
.................... 52
3.4
The key lemma
........................ 52
3.5
Chevalley s Theorem
..................... 54
3.6
The module of
covariante
.................. 56
3.7
Uniqueness of the degrees
.................. 58
3.8
Eigenvalues
.......................... 60
3.9
Sum and product of the degrees
............... 62
3.10
Jacobian criterion for algebraic independence
....... 63
3.11
Groups with free rings of invariants
............. 65
3.12
Examples
........................... 66
3.13
Factorization of the Jacobian
................ 68
3.14
Induction and restriction of class functions
......... 70
3.15
Factorization of the
Poincaré
polynomial
.......... 71
3.16
Coxeter elements
....................... 74
3.17
Action on a plane
....................... 76
3.18
The Coxeter number
..................... 79
3.19
Eigenvalues of Coxeter elements
............... 80
3.20
Exponents and degrees of Weyl groups
........... 82
Affine
reflection groups
87
4.1
Affine
reflections
....................... 87
4.2
Affine Weyl
groups
...................... 88
4.3
Alcoves
............................ 89
4.4
Counting
hyperplanes.................... 91
4.5
Simple transitivity
...................... 92
4.6
Exchange Condition
..................... 94
4.7
Coxeter graphs and extended Dynkin diagrams
...... 95
4.8
Fundamental domain
..................... 96
4.9
A formula for the order of
W
................ 97
4.10
Groups generated by
affine
reflections
........... 99
їх
II
General
theory of Coxeter groups
103
5
Coxeter groups
105
5.1
Coxeter systems
.......................105
5.2
Length function
........................107
5.3
Geometric representation of
W
...............108
5.4
Positive
ала
negative roots
.................
Ill
5.5
Parabolic subgroups
.....................113
5.6
Geometric interpretation of the length function
......114
5.7
Roots and reflections
.....................116
5.8
Strong Exchange Condition
.................117
5.9
Bruhat ordering
........................118
5.10
Subexpressions
........................120
5.11
Intervals in the Bruhat ordering
...............121
5.12
Poincaré
series
........................122
5.13
Fundamental domain for
W
.................124
6
Special cases
129
6.1
Irreducible Coxeter systems
.................129
6.2
More on the geometric representation
...........130
6.3
Radical of the bilinear form
.................131
6.4
Finite Coxeter groups
....................132
6.5 Affine
Coxeter groups
....................133
6.6
Crystallographic Coxeter groups
..............135
6.7
Coxeter groups of rank
3...................137
6.8
Hyperbolic Coxeter groups
..................138
6.9
List of hyperbolic Coxeter groups
..............141
7 Hecke
algebras and Kazhdan-Lusztig polynomials
145
7.1
Generic algebras
.........,..............145
7.2
Commuting operators
....................147
7.3
Conclusion of the proof
...................149
7.4 Hecke
algebras and inverses
.................150
7.5
Computing the
Л
-polynomials................
152
7.6
Special case: finite Coxeter groups
.............154
7.7
An involution on
H
......................155
7.8
Further properties of
fí-polynomials
............156
7.9
Kazhdan-Lusztig polynomials
................157
7.10
Uniqueness
..........................159
7.11
Existence
...........................160
7.12
Examples
...........................162
7.13
Inverse Kazhdan-Lusztig polynomials
...........164
7.14
Multiplication formulas
...................166
7.15
Cells and representations of
Hecke
algebras
........167
x
Contents
8
Complements
171
8.1
The Word Problem
......................171
8.2
Reflection subgroups
.....................172
8.3
Involutions
..........................173
8.4
Coxeter elements and their eigenvalues
...........174
8.5
Möbius
function of the Bruhat ordering
..........175
8.6
Intervals and Bruhat graphs
.................176
8.7
Shellability
..........................177
8.8
Automorphisms of the Bruhat ordering
...........178
8.9
Poincaré
series of
affine Weyl
groups
............179
8.10
Representations of finite Coxeter
groups
.............................180
8.11 Schur
multipliers
.......................181
8.12
Coxeter groups and Lie theory
...............182
References
185
Index
203
|
| any_adam_object | 1 |
| author | Humphreys, James E. 1939-2020 |
| author_GND | (DE-588)108120848 |
| author_facet | Humphreys, James E. 1939-2020 |
| author_role | aut |
| author_sort | Humphreys, James E. 1939-2020 |
| author_variant | j e h je jeh |
| building | Verbundindex |
| bvnumber | BV035309816 |
| classification_rvk | SK 260 SK 340 |
| classification_tum | MAT 200f |
| ctrlnum | (OCoLC)318263113 (DE-599)BVBBV035309816 |
| dewey-full | 512.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.2 |
| dewey-search | 512.2 |
| dewey-sort | 3512.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. paperback ed. (with corr.), transferred to digital print. |
| format | Book |
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| indexdate | 2024-07-09T21:30:58Z |
| institution | BVB |
| isbn | 0521436133 9780521436137 |
| language | English |
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| physical | XII, 204 S. graph. Darst. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Cambridge Univ. Pr. |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Humphreys, James E. 1939-2020 Verfasser (DE-588)108120848 aut Reflection groups and Coxeter groups James E. Humphreys 1. paperback ed. (with corr.), transferred to digital print. Cambridge u.a. Cambridge Univ. Pr. 2000 XII, 204 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 29 Literaturverz. S. 185 - 201 Grupos de Coxeter Grupos finitos Coxeter-Gruppe (DE-588)4261522-7 gnd rswk-swf Reflexionsgruppe gnd rswk-swf Coxeter-Gruppe (DE-588)4261522-7 s DE-604 Reflexionsgruppe f Cambridge studies in advanced mathematics 29 (DE-604)BV000003678 29 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017114579&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Humphreys, James E. 1939-2020 Reflection groups and Coxeter groups Cambridge studies in advanced mathematics Grupos de Coxeter Grupos finitos Coxeter-Gruppe (DE-588)4261522-7 gnd |
| subject_GND | (DE-588)4261522-7 |
| title | Reflection groups and Coxeter groups |
| title_auth | Reflection groups and Coxeter groups |
| title_exact_search | Reflection groups and Coxeter groups |
| title_full | Reflection groups and Coxeter groups James E. Humphreys |
| title_fullStr | Reflection groups and Coxeter groups James E. Humphreys |
| title_full_unstemmed | Reflection groups and Coxeter groups James E. Humphreys |
| title_short | Reflection groups and Coxeter groups |
| title_sort | reflection groups and coxeter groups |
| topic | Grupos de Coxeter Grupos finitos Coxeter-Gruppe (DE-588)4261522-7 gnd |
| topic_facet | Grupos de Coxeter Grupos finitos Coxeter-Gruppe Reflexionsgruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017114579&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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