Reflection Groups and Invariant Theory

Transcription

1 Richard Kane Reflection Groups and Invariant Theory Springer

2 Introduction 1 Reflection groups 5 1 Euclidean reflection groups Reflections and reflection groups Groups of symmetries in the plane Dihedral groups Planar reflection groups as dihedral groups Groups of symmetries in 3-space Weyl chambers Invariant theory 21 2 Root systems Root systems Examples of root systems Crystallographic root systems Essential root systems and stable isomorphisms 32 3 Fundamental systems Fundamental systems Examples of fundamental systems Existence of fundamental systems Fundamental systems and positive roots Weyl chambers and fundamental systems Height 43 4 Length Fundamental reflections Length Length and root systems Matsumoto cancellation Length and reflecting hyperplanes The action of W(A) on fundamental systems and Weyl chambers 54 5 Parabolic subgroups* Parabolic subgroups Isotropy subgroups Conjugation of parabolic subgroups 63

3 vj II Coxeter groups 65 6 Reflection groups and Coxeter systems Coxeter groups and Coxeter systems Reflection groups are Coxeter groups The uniqueness of Coxeter structures 72 7 Bilinear forms of Coxeter systems The bilinear form of a Coxeter system The Tits representation Positive definiteness 78 8 Classification of Coxeter systems and reflection groups Classification results Preliminary results The two possible cases The chain case The ramification case Coxeter graphs of root systems 92 III Weyl groups 97 9 Weyl groups Weyl groups The root lattice Q Coroots and the coroot lattice Q v Fundamental weights and the weight lattice"? Equivariant lattices The Classification of crystallographic root systems Isomorphism of root systems Cartan matrices Ill 10-3 Angles and ratios of lengths Coxeter graphs and Dynkin diagrams The classification of root systems Affine Weyl groups The affine Weyl group The highest root Affine Weyl groups as Coxeter groups Affine root systems Alcoves The order of Weyl groups Subroot systems The Borel-de Siebenthal theorem The subroot system A(f) Maximal subroot systems Characterizations of the root systems A(t) Maximal root systems Formal identities The MacDonald identity The element p The element* The Weyl identity 148

4 vii 13-5 The proof of the MacDonald identity The proof of polynomial identity 150 IV Pseudo-reflection groups Pseudo-reflections F (Generalized) reflections Pseudo-reflections The modular and nonmodular cases Classifications of pseudo-reflection groups Complex pseudo-reflection groups Other pseudo-reflection groups in characteristic Pseudo-reflection groups in characteristic p 165 V Rings of invariants The ring of invariants The ring of invariants Examples Extension theory Properties of rings of invariants The Dickson invariants Poincare series Poincare series Molien's theorem Molien's theorem and pseudo-reflections Polynomial algebras as rings of invariants The algebra of covariants Nonmodular invariants of pseudo-reflection groups The main result The A operators S as a free R module R as a polynomial algebra G as a pseudo-reflection group 199 _ Invariants of Euclidean reflection groups Modular invariants of pseudo-reflection groups Polynomial rings of invariants Generalized invariants Regular sequences 209 VI Skew invariants Skew invariants Skew invariants The element Q The ring of covariants Thejacobian Thejacobian The proof of Proposition A The proof of Proposition B Extended partial derivatives The chain rule 226

5 viii 22 The extended ring of invariants Exterior algebras The differential d: S(V) E(V) -* S(V) <B> E(V) Invariants of S(V) E(V) The Poincare series of [S(V) E{V)] G 234 VII Rings of covariants Poincare series for the ring of covariants Poincare series The exponents of Weyl groups The A operations The element u> The proof of Proposition Representations of pseudo-reflection groups SG as the regular representation The Poincare series of irreducible representations Exterior powers of reflection representation MacDonald representations Harmonic elements Hopf algebras Differential operators Group actions Harmonic elements Harmonics and reflection groups Main results Harmonics of pseudo-reflection groups are cyclic Generalized harmonics Cyclic harmonics Pseudo-reflection groups are characterized via harmonics Isotropy subgroups Poincare duality 278 VIII Conjugacy classes Involutions Elements of greatest length The involution c = The involutions c Conjugacy classes of involutions Conjugacy classes and Coxeter graphs Elementary equivalences Summary Equivalences via Coxeter graph symmetries Elementary equivalences Decomposition of W equivalences into elementary equivalences Involutions Coxeter elements Coxeter elements 299

6 ix 29-2 Coxeter elements are conjugate A dihedral subgroup The order of Coxeter elements Centralizers of Coxeter elements Regular elements Minimal decompositions Main results TheproofofPropositions30-lAand30-lB The proof of Theorem 30-1A The proof of Theorem 30-IB 317 IX Eigenvalues Eigenvalues for reflection groups Eigenspaces and exponents The proof of Theorem A The proof of Theorem B Eigenvalues for regular elements Regular elements Eigenvalues of regular elements The proof of Theorem 32-1A Eigenvalues in Euclidean reflection groups Ring of invariants and eigenvalues Main results Algebraic geometry The ring of invariants as a coordinate ring Eigenspaces Properties of regular elements Properties of regular elements Conjugacy classes of regular elements Conjugacy classes of regular elements in Weyl groups 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