Richard Kane Reflection Groups and Invariant Theory Springer
Introduction 1 Reflection groups 5 1 Euclidean reflection groups 6 1-1 Reflections and reflection groups 6 1-2 Groups of symmetries in the plane 8 1-3 Dihedral groups 9 1-4 Planar reflection groups as dihedral groups 12 1-5 Groups of symmetries in 3-space 14 1-6 Weyl chambers 18 1-7 Invariant theory 21 2 Root systems 25 2-1 Root systems 25 2-2 Examples of root systems 29 2-3 Crystallographic root systems 31 2-4 Essential root systems and stable isomorphisms 32 3 Fundamental systems 35 3-1 Fundamental systems 35 3-2 Examples of fundamental systems 36 3-3 Existence of fundamental systems 37 3-4 Fundamental systems and positive roots 40 3-5 Weyl chambers and fundamental systems 41 3-6 Height 43 4 Length 45 4-1 Fundamental reflections 45 4-2 Length 46 4-3 Length and root systems 47 4-4 Matsumoto cancellation 51 4-5 Length and reflecting hyperplanes 53 4-6 The action of W(A) on fundamental systems and Weyl chambers 54 5 Parabolic subgroups* 57 5-1 Parabolic subgroups 57 5-2 Isotropy subgroups 60 5-3 Conjugation of parabolic subgroups 63
vj II Coxeter groups 65 6 Reflection groups and Coxeter systems 66 6-1 Coxeter groups and Coxeter systems 66 6-2 Reflection groups are Coxeter groups 68 6-3 The uniqueness of Coxeter structures 72 7 Bilinear forms of Coxeter systems 75 7-1 The bilinear form of a Coxeter system 75 7-2 The Tits representation 76 7-3 Positive definiteness 78 8 Classification of Coxeter systems and reflection groups 81 8-1 Classification results 81 8-2 Preliminary results 83 8-3 The two possible cases 85 8-4 The chain case 87 8-5 The ramification case 89 8-6 Coxeter graphs of root systems 92 III Weyl groups 97 9 Weyl groups 98 9-1 Weyl groups 98 9-2 The root lattice Q 100 9-3 Coroots and the coroot lattice Q v 101 9-4 Fundamental weights and the weight lattice"? 103 9-5 Equivariant lattices 105 10 The Classification of crystallographic root systems 109 10-1 Isomorphism of root systems 109 10-2 Cartan matrices Ill 10-3 Angles and ratios of lengths 114 10-4 Coxeter graphs and Dynkin diagrams 115 10-5 The classification of root systems 116 11 Affine Weyl groups 118 11-1 The affine Weyl group 118 11-2 The highest root 119 11-3 Affine Weyl groups as Coxeter groups 123 11-4 Affine root systems 125 11-5 Alcoves 129 11-6 The order of Weyl groups 131 12 Subroot systems 135 12-1 The Borel-de Siebenthal theorem 135 12-2 The subroot system A(f) 138 12-3 Maximal subroot systems 138 12-4 Characterizations of the root systems A(t) 139 12-5 Maximal root systems 141 13 Formal identities 144 13-1 The MacDonald identity 144 13-2 The element p. 145 13-3 The element* 147 13-4 The Weyl identity 148
vii 13-5 The proof of the MacDonald identity 150 13-6 The proof of polynomial identity 150 IV Pseudo-reflection groups 153 14 Pseudo-reflections 154 14-F (Generalized) reflections 154 14-2 Pseudo-reflections 157 14-3 The modular and nonmodular cases 158 15 Classifications of pseudo-reflection groups 161 15-1 Complex pseudo-reflection groups 161 15-2 Other pseudo-reflection groups in characteristic 0... 164 15-3 Pseudo-reflection groups in characteristic p 165 V Rings of invariants 169 16 The ring of invariants 170 16-1 The ring of invariants 170 16-2 Examples 172 16-3 Extension theory 173 16-4 Properties of rings of invariants 175 16-5 The Dickson invariants 177 17 Poincare series 180 17-1 Poincare series 180 17-2 Molien's theorem 181 17-3 Molien's theorem and pseudo-reflections 185 17-4 Polynomial algebras as rings of invariants 187 17-5 The algebra of covariants 188 18 Nonmodular invariants of pseudo-reflection groups 191 18-1 The main result 191 18-2 The A operators 192 18-3 S as a free R module 194 18-4 R as a polynomial algebra 195 18-5 G as a pseudo-reflection group 199 _. 18-6 Invariants of Euclidean reflection groups 200 19 Modular invariants of pseudo-reflection groups 202 19-1 Polynomial rings of invariants 202 19-2 Generalized invariants 207 19-3 Regular sequences 209 VI Skew invariants 213 20 Skew invariants 214 20-1 Skew invariants 214 20-2 The element Q. 215 20-3 The ring of covariants... 219 21 Thejacobian 221 21-1 Thejacobian 221 21-2 The proof of Proposition A 222 21-3 The proof of Proposition B 223 21-4 Extended partial derivatives 224 21-5 The chain rule 226
viii 22 The extended ring of invariants 229 22-1 Exterior algebras 229 22-2 The differential d: S(V) E(V) -* S(V) <B> E(V) 230 22-3 Invariants of S(V) E(V) 232 22-4 The Poincare series of [S(V) E{V)] G 234 VII Rings of covariants 235 23 Poincare series for the ring of covariants 236 23-1 Poincare series 236 23-2 The exponents of Weyl groups 237 23-3 The A operations 239 23-4 The element u> 0 241 23-5 The proof of Proposition 23-3 243 24 Representations of pseudo-reflection groups 247 24-1 SG as the regular representation 247 24-2 The Poincare series of irreducible representations... 248 24-3 Exterior powers of reflection representation 250 24-4 MacDonald representations 254 25 Harmonic elements 256 25-1 Hopf algebras 256 25-2 Differential operators 258 25-3 Group actions 260 25-4 Harmonic elements 261 26 Harmonics and reflection groups 263 26-1 Main results 263 26-2 Harmonics of pseudo-reflection groups are cyclic... 264 26-3 Generalized harmonics 267 26-4 Cyclic harmonics 270 26-5 Pseudo-reflection groups are characterized via harmonics 274 26-6 Isotropy subgroups 277 26-7 Poincare duality 278 VIII Conjugacy classes 279 27 Involutions 280 27-1 Elements of greatest length 280 27-2 The involution c = -1 283 27-3 The involutions c 7 283 27-4 Conjugacy classes of involutions 286 27-5 Conjugacy classes and Coxeter graphs 288 28 Elementary equivalences 290 28-1 Summary 290 28-2 Equivalences via Coxeter graph symmetries 291 28-3 Elementary equivalences 292 28-4 Decomposition of W equivalences into elementary equivalences 293 28-5 Involutions 295 29 Coxeter elements 299 29-1 Coxeter elements 299
ix 29-2 Coxeter elements are conjugate 300 29-3 A dihedral subgroup 302 29-4 The order of Coxeter elements 306 29-5 Centralizers of Coxeter elements 307 29-6 Regular elements 309 30 Minimal decompositions 311 30-1 Main results 311 30-2 TheproofofPropositions30-lAand30-lB 313 30-3 The proof of Theorem 30-1A 315 30-4 The proof of Theorem 30-IB 317 IX Eigenvalues 319 31 Eigenvalues for reflection groups 320 31-1 Eigenspaces and exponents 320 31-2 The proof of Theorem A 322 31-3 The proof of Theorem B 323 32 Eigenvalues for regular elements 325 32-1 Regular elements 325 32-2 Eigenvalues of regular elements 327 32-3 The proof of Theorem 32-1A 329 32-4 Eigenvalues in Euclidean reflection groups 330 33 Ring of invariants and eigenvalues 334 33-1 Main results 334 33-2 Algebraic geometry 335 33-3 The ring of invariants as a coordinate ring 336 33-4 Eigenspaces 338 34 Properties of regular elements 341 34-1 Properties of regular elements 341 34-2 Conjugacy classes of regular elements 343 34-3 Conjugacy classes of regular elements in Weyl groups.. 345 34-4 Centralizers of regular elements 346 Appendices A Rings and modules 350 B Group actions and representation theory 354 C Quadratic forms 361 D Lie algebras 366 References 369 Index 377