## Reflection Groups and Coxeter GroupsIn 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 |

203 | |

### Common terms and phrases

action acts algebra apply arbitrary associated assume basis Bruhat ordering called cells Chapter clear coefficient combination complex compute Condition connected consider consisting contains Corollary corresponding Coxeter graph Coxeter groups crystallographic define Denote determinant eigenvalues elements equal example Exercise exists fact factors finite finite reflection groups fixed follows forcing formula function geometric given group of type hyperplanes implies induction intersection invariants involving irreducible l(sw labelled lattice Lemma length length function lies linear matrix multiplication natural negative nonzero Note obtain occurs orthogonal pair particular permutation polynomials positive positive definite possible precisely Proof Proposition prove Recall reduced expression relations relative representation result ring root system satisfying shows side simple system space span subgroup subset Suppose symmetric Theorem theory turn unique vectors Weyl groups write yields

### Popular passages

Page 187 - ... 802. 00036 Carrato, S. linage vector quantization using ordered codebooks Properties and applications Signal Process 40, No I, 87-103 (1994). 812. 68117 Carrcga, Jean-Claude Internal Boolean embedding for distributive sublaliices Acta Sei Math. 59 No 1-2 5V59 (1994) 803. 06012 Carrell, James B. Some remarks on regular Weyl group orbits and the cohomology of Schubert varieties In Dcodhar. Vinay (cd ) Kazhdan-Lusztig theory and related topics Proceedings of an AMS special session, held May 19-20....

Page 191 - Invariants of finite reflection groups and mean value problems, Amer. J. Math. 91 (1969), 591-598.

Page 196 - ON THE DEGREES OF THE IRREDUCIBLE REPRESENTATIONS OF FINITE COXETER GROUPS. J. LONDON MATH. SOC., II.

Page 192 - Gyoja, A., A generalized Poincare series associated to a Hecke algebra of a finite or p-adic Chevalley group, Japan, J. Math., 9 87-111 (1983).

Page 202 - An approach to the connectedness of the left cells in affine Weyl groups, Bull.