Reflection Groups and Coxeter Groups

Cambridge University Press, 1992 - Mathematics - 204 pages
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 Copyright

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