Reflection Groups and Invariant Theory

Couverture
Springer Science & Business Media, 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.
 

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Table des matières

I Reflection groups
5
Euclidean reflection groups
6
12 Groups of symmetries in the plane
8
13 Dihedral groups
9
14 Planar reflection groups as dihedral groups
12
15 Groups of symmetries in 3space
14
17 Invariant theory
21
Root systems
25
184 R as a polynomial algebra
195
185 G as a pseudoreflection group
199
186 Invariants of Euclidean reflection groups
200
Modular invariants of pseudoreflection groups
202
192 Generalized invariants
207
193 Regular sequences
209
Skew invariants
213
Skew invariants
214

22 Examples of root systems
29
23 Crystallographic root systems
31
Fundamental systems
35
32 Examples of fundamental systems
36
33 Existence of fundamental systems
37
34 Fundamental systems and positive roots
40
35 Weyl chambers and fundamental systems
41
36 Height
43
Length
45
42 Length
46
43 Length and root systems
47
44 Matsumoto cancellation
51
Parabolic subgroups
57
52 Isotropy subgroups
60
53 Conjugation of parabolic subgroups
63
Coxeter groups
65
Reflection groups and Coxeter systems
66
62 Reflection groups are Coxeter groups
68
63 The uniqueness of Coxeter structures
72
Bilinear forms of Coxeter systems
75
72 The Tits representation
76
73 Positive definiteness
78
Classification of Coxeter systems and reflection groups
81
82 Preliminary results
83
83 The two possible cases
85
84 The chain case
87
85 The ramification case
89
86 Coxeter graphs of root systems
92
Weyl groups
97
Weyl groups
98
92 The root lattice Q
100
93 Coroots and the coroot lattice Qv
101
94 Fundamental weights and the weight lattice P
103
95 Equivariant lattices
105
The Classification of crystallographic root systems
109
102 Cartan matrices
111
103 Angles and ratios of lengths
114
104 Coxeter graphs and Dynkin diagrams
115
105 The classification of root systems
116
Affine Weyl groups
118
112 The highest root
119
113 Affine Weyl groups as Coxeter groups
123
114 Affine root systems
125
115 Alcoves
129
116 The order of Weyl groups
131
Subroot systems
135
122 The subroot system Af
138
124 Characterizations of the root systems Af
139
125 Maximal root systems
141
Formal identities
144
132 The element p
145
134 The Weyl identity
148
135 The proof of the MacDonald identity
150
Pseudoreflection groups
153
Pseudoreflections
154
142 Pseudoreflections
157
143 The modular and nonmodular cases
158
Classifications of pseudoreflection groups
161
152 Other pseudoreflection groups in characteristic 0
164
153 Pseudoreflection groups in characteristic p
165
Rings of invariants
169
The ring of invariants
170
162 Examples
172
163 Extension theory
173
165 The Dickson invariants
177
Poincare series
180
172 Moliens theorem
181
174 Polynomial algebras as rings of invariants
187
Nonmodular invariants of pseudoreflection groups
191
183 5 as a free R module
194
202 The element fi
215
203 The ring of covariants
219
The Jacobian
221
212 The proof of Proposition A
222
213 The proof of Proposition B
223
214 Extended partial derivatives
224
215 The chain rule
226
The extended ring of invariants
229
SV EV SV EEV The map d V F EV extends to a differential map
230
223 Invariants of SV ii V
232
224 The Poincare series of SV E EVG
234
Rings of covariants
235
Poincare series for the ring of covariants
236
232 The exponents of Weyl groups
237
233 The A operations
239
234 The element u0
241
Representations of pseudoreflection groups
247
242 The Poincare series of irreducible representations
248
243 Exterior powers of reflection representation
250
244 MacDonald representations
254
Harmonic elements
256
252 Differential operators
258
253 Group actions
260
254 Harmonic elements
261
Harmonics and reflection groups
263
263 Generalized harmonics
267
264 Cyclic harmonics
270
265 Pseudoreflection groups are characterized via harmonics
274
267 Poincare duality
278
Conjugacy classes
279
Involutions
280
272 The involution c 1
283
274 Conjugacy classes of involutions
286
275 Conjugacy classes and Coxeter graphs
288
Elementary equivalences
290
282 Equivalences via Coxeter graph symmetries
291
283 Elementary equivalences
292
284 Decomposition of W equivalences into elementary equivalences
293
285 Involutions
295
Coxeter elements
299
292 Coxeter elements are conjugate
300
293 A dihedral subgroup
302
294 The order of Coxeter elements
306
295 Centralizers of Coxeter elements
307
296 Regular elements
309
Minimal decompositions
311
302 The proof of Propositions 301A and 301B
313
303 The proof of Theorem 301A
315
Eigenvalues
319
Eigenvalues for reflection groups
320
312 The proof of Theorem A
322
313 The proof of Theorem B
323
Eigenvalues for regular elements
325
322 Eigenvalues of regular elements
327
323 The proof of Theorem 321A
329
324 Eigenvalues in Euclidean reflection groups
330
Ring of invariants and eigenvalues
334
332 Algebraic geometry
335
333 The ring of invariants as a coordinate ring
336
334 Eigenspaces
338
Properties of regular elements
341
343 Conjugacy classes of regular elements in Weyl groups
345
344 Centralizers of regular elements
346
A Rings and modules
350
Group actions and representation theory
354
Quadratic forms
361
Lie algebras
366
References
369
Index
377
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