Reflection Groups and Invariant Theory

Couverture
Springer Science & Business Media, 21 juin 2001 - 379 pages
Reflection Groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra.
Richard Kane is a professor of mathematics at the University of Western Ontario. His research interests are algebra and algebraic topology. Professor Kane is a former President of the Canadian Mathematical Society.
 

Table des matières

I Reflection groups
Euclidean reflection groups
12 Groups of symmetries in the plane
13 Dihedral groups
15 Groups of symmetries in 3space
17 Invariant theory
1
Root systems
5
22 Examples of root systems
9
185 G as a pseudoreflection group
137
186 Invariants of Euclidean reflection groups
137
Modular invariants of pseudoreflection groups
137
192 Generalized invariants
137
193 Regular sequences
137
Skew invariants
137
Skew invariants
137
202 The element fi
137

23 Crystallographic root systems
11
Fundamental systems
15
32 Examples of fundamental systems
16
33 Existence of fundamental systems
17
34 Fundamental systems and positive roots
20
35 Weyl chambers and fundamental systems
21
36 Height
23
Length
25
42 Length
26
43 Length and root systems
27
44 Matsumoto cancellation
27
Parabolic subgroups
27
52 Isotropy subgroups
27
53 Conjugation of parabolic subgroups
29
Coxeter groups
31
Reflection groups and Coxeter systems
32
62 Reflection groups are Coxeter groups
34
63 The uniqueness of Coxeter structures
38
Bilinear forms of Coxeter systems
41
72 The Tits representation
42
73 Positive definiteness
44
Classification of Coxeter systems and reflection groups
45
82 Preliminary results
45
83 The two possible cases
45
84 The chain case
47
85 The ramification case
49
86 Coxeter graphs of root systems
52
Weyl groups
53
Weyl groups
53
92 The root lattice Q
54
93 Coroots and the coroot lattice Qv
55
94 Fundamental weights and the weight lattice P
57
95 Equivariant lattices
59
The Classification of crystallographic root systems
61
102 Cartan matrices
61
103 Angles and ratios of lengths
62
104 Coxeter graphs and Dynkin diagrams
63
Affine Weyl groups
66
112 The highest root
67
113 Affine Weyl groups as Coxeter groups
69
114 Affine root systems
71
115 Alcoves
75
116 The order of Weyl groups
75
Subroot systems
79
122 The subroot system Af
82
124 Characterizations of the root systems Af
83
125 Maximal root systems
85
Formal identities
88
132 The element p
89
134 The Weyl identity
92
135 The proof of the MacDonald identity
94
Pseudoreflection groups
97
Pseudoreflections
98
142 Pseudoreflections
101
143 The modular and nonmodular cases
102
Classifications of pseudoreflection groups
105
152 Other pseudoreflection groups in characteristic 0
108
153 Pseudoreflection groups in characteristic p
109
Rings of invariants
113
The ring of invariants
114
162 Examples
116
163 Extension theory
117
165 The Dickson invariants
121
Poincare series
121
172 Moliens theorem
121
174 Polynomial algebras as rings of invariants
125
Nonmodular invariants of pseudoreflection groups
129
183 5 as a free R module
132
184 R as a polynomial algebra
133
203 The ring of covariants
139
The Jacobian
141
212 The proof of Proposition A
141
213 The proof of Proposition B
141
214 Extended partial derivatives
141
215 The chain rule
141
The extended ring of invariants
143
SV EV SV EEV The map d V F EV extends to a differential map
144
223 Invariants of SV ii V
146
224 The Poincare series of SV E EVG
148
Rings of covariants
149
Poincare series for the ring of covariants
150
232 The exponents of Weyl groups
151
233 The A operations
153
234 The element u0
155
Representations of pseudoreflection groups
161
242 The Poincare series of irreducible representations
162
243 Exterior powers of reflection representation
164
Harmonic elements
170
252 Differential operators
171
253 Group actions
172
254 Harmonic elements
173
Harmonics and reflection groups
175
263 Generalized harmonics
179
264 Cyclic harmonics
182
265 Pseudoreflection groups are characterized via harmonics
186
267 Poincare duality
190
Conjugacy classes
191
Involutions
192
272 The involution c 1
195
274 Conjugacy classes of involutions
198
275 Conjugacy classes and Coxeter graphs
200
Elementary equivalences
201
282 Equivalences via Coxeter graph symmetries
201
283 Elementary equivalences
202
284 Decomposition of W equivalences into elementary equivalences
203
285 Involutions
203
Coxeter elements
207
292 Coxeter elements are conjugate
208
293 A dihedral subgroup
210
294 The order of Coxeter elements
214
295 Centralizers of Coxeter elements
215
296 Regular elements
217
Minimal decompositions
219
302 The proof of Propositions 301A and 301B
221
303 The proof of Theorem 301A
223
Eigenvalues
225
Eigenvalues for reflection groups
225
312 The proof of Theorem A
225
313 The proof of Theorem B
225
Eigenvalues for regular elements
227
322 Eigenvalues of regular elements
227
323 The proof of Theorem 321A
227
324 Eigenvalues in Euclidean reflection groups
227
Ring of invariants and eigenvalues
227
332 Algebraic geometry
227
333 The ring of invariants as a coordinate ring
227
334 Eigenspaces
228
Properties of regular elements
231
343 Conjugacy classes of regular elements in Weyl groups
234
344 Centralizers of regular elements
234
A Rings and modules
234
Group actions and representation theory
234
Quadratic forms
234
Lie algebras
238
References
241
Index
249
Droits d'auteur

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Expressions et termes fréquents

A₁ action of G affine Weyl group alcoves associated bilinear form chamber Chapter char F closed subroot systems coefficients conjugacy classes conjugate Coxeter elements Coxeter graph Coxeter group Coxeter systems crystallographic root systems d₁ decomposed decomposition defined degree dihedral group Dynkin diagram eigenspaces eigenvalue Euclidean reflection group example F vector fact finite Euclidean reflection Finite Groups finite reflection groups follows fundamental system given graded F algebra group G C GL(V hyperplane ideal identity invariant theory involutions irreducible representations isomorphism lattice Lemma linear Math maximal minimal multiplication Noetherian obtain orthogonal parabolic subgroups permutes Poincaré series polynomial algebra primitive d-th root proof of Proposition prove pseudo-reflection groups reflecting hyperplanes regular elements relation representation of G result ring of invariants root of unity root system skew invariant subspace t₁ vector space W₁ Waff

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Informations bibliographiques

TitreReflection Groups and Invariant Theory
CMS Books in Mathematics, Canadian Mathematical Society, ISSN 1613-5237
Canadian Mathematical Society books in mathematics
AuteurRichard Kane
Éditionillustrée
ÉditeurSpringer Science & Business Media, 2001
ISBN038798979X, 9780387989792
Longueur379 pages
  
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