Reflection Groups and Coxeter Groups

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Cambridge University Press, 1992 - Mathematics - 204 pages
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In this graduate textbook Professor Humphreys presents a concrete and up-to-date introduction to the theory of Coxeter groups. He assumes that the reader has a good knowledge of algebra, but otherwise the book is self contained. The first part is devoted to establishing concrete examples; the author begins by developing the most important facts about finite reflection groups and related geometry, and showing that such groups have a Coxeter representation. In the next chapter these groups are classified by Coxeter diagrams, and actual realizations of these groups are discussed. Chapter 3 discusses the polynomial invariants of finite reflection groups, and the first part ends with a description of the affine Weyl groups and the way they arise in Lie theory. The second part (which is logically independent of, but motivated by, the first) starts by developing the properties of the Coxeter groups. Chapter 6 shows how earlier examples and others fit into the general classification of Coxeter diagrams. Chapter 7 is based on the very important work of Kazhdan and Lusztig and the last chapter presents a number of miscellaneous topics of a combinatorial nature.
 

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Contents

Finite reflection groups
3
12 Roots
6
13 Positive and simple systems
7
14 Conjugacy of positive and simple systems
10
16 The length function
12
17 Deletion and Exchange Conditions
13
18 Simple transitivity and the longest element
15
19 Generators and relations
16
410 Groups generated by affine reflections
99
General theory of Coxeter groups
103
Coxeter groups
105
52 Length function
107
53 Geometric representation of W
108
54 Positive and negative roots
111
55 Parabolic subgroups
113
56 Geometric interpretation of the length function
114

110 Parabolic subgroups and minimal coset representatives
18
111 Poincare polynomials
20
112 Fundamental domains
21
113 The lattice of parabolic subgroups
24
115 The Coxeter complex
25
116 An alternating sum formula
26
Classification of finite reflection groups
29
22 Irreducible components
30
23 Coxeter graphs and associated bilinear forms
31
24 Some positive definite graphs
32
25 Some positive semidefinite graphs
33
26 Subgraphs
35
27 Classification of graphs of positive type
36
28 Crystallographic groups
38
29 Crystallographic root systems and Weyl groups
39
210 Construction of root systems
41
211 Computing the order of W
43
212 Exceptional Weyl groups
45
213 Groups of types H₃ and H₄
46
Polynomial invariants of finite reflection groups
49
32 Finite generation
50
33 A divisibility criterion
52
35 Chevalleys Theorem
54
36 The module of covariants
56
37 Uniqueness of the degrees
58
38 Eigenvalues
60
39 Sum and product of the degrees
62
310 Jacobian criterion for algebraic independence
63
311 Groups with free rings of invariants
65
312 Examples
66
313 Factorization of the Jacobian
68
314 Induction and restriction of class functions
70
315 Factorization of the Poincare polynomial
71
316 Coxeter elements
74
317 Action on a plane
76
318 The Coxeter number
79
319 Eigenvalues of Coxeter elements
80
320 Exponents and degrees of Weyl groups
82
Affine reflection groups
87
42 Affine Weyl groups
88
43 Alcoves
89
44 Counting hyperplanes
91
45 Simple transitivity
92
46 Exchange Condition
94
47 Coxeter graphs and extended Dynkin diagrams
95
48 Fundamental domain
96
49 A formula for the order of W
97
57 Roots and reflections
116
58 Strong Exchange Condition
117
59 Bruhat ordering
118
510 Subexpressions
120
511 Intervals in the Bruhat ordering
121
512 Poincare series
122
513 Fundamental domain for W
124
Special cases
129
62 More on the geometric representation
130
63 Radical of the bilinear form
131
64 Finite Coxeter groups
132
65 Affine Coxeter groups
133
66 Crystallographic Coxeter groups
135
67 Coxeter groups of rank 3
137
68 Hyperbolic Coxeter groups
138
69 List of hyperbolic Coxeter groups
141
Hecke algebras and KazhdanLusztig polynomials
145
72 Commuting operators
147
73 Conclusion of the proof
149
74 Hecke algebras and inverses
150
75 Computing the R polynomials
152
finite Coxeter groups
154
77 An involution on 𝓗
155
78 Further properties of Rpolynomials
156
79 KazhdanLusztig polynomials
157
710 Uniqueness
159
711 Existence
160
712 Examples
162
713 Inverse Kazhdan Lusztig polynomials
164
714 Multiplication formulas
166
715 Cells and representations of Hecke algebras
167
Complements
171
82 Reflection subgroups
172
83 Involutions
173
84 Coxeter elements and their eigenvalues
174
85 Mobius function of the Bruhat ordering
175
86 Intervals and Bruhat graphs
176
87 Shellability
177
88 Automorphisms of the Bruhat ordering
178
89 Poincare series of affine Weyl groups
179
810 Representations of finite Coxeter groups
180
811 Schur multipliers
181
812 Coxeter groups and Lie theory
182
References
185
Index
203
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Page 196 - ON THE DEGREES OF THE IRREDUCIBLE REPRESENTATIONS OF FINITE COXETER GROUPS. J. LONDON MATH. SOC., II.
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About the author (1992)

James E. Humphreys was born in Erie, Pennsylvania, and received his A.B. from Oberlin College, 1961, and his Ph.D. from Yale University, 1966. He has taught at the University of Oregon, Courant Institute (NYU), and the University of Massachusetts at Amherst (now retired). He visits IAS Princeton, Rutgers. He is the author of several graduate texts and monographs.

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