Coxeter element

Abstract: I.- The basics.- Bruhat order.- Weak order and reduced words.- Roots, games, and automata.- II.- Kazhdan-Lusztig and R-polynomials.- Kazhdan-Lusztig representations.- Enumeration.- Combinatorial Descriptions.

Abstract: here l(w) is the length of w In the case where Wis a Weyl group and q is specialized to a fixed prime power, | ~ can be interpreted as the algebra of intertwining operators of the space of functions on the flag manifold of the corresponding finite Chevalley group G(Fq) (see [loc cit, Ex 24]) Therefore, the problem of decomposing this space of functions into irreducible representations of G(Fq) is equivalent to the problem of decomposing the regular representation o f ~ | (12 It is known that, in this case, | is isomorphic to the group algebra of W; however, in general, this isomorphism cannot be defined without introducing a square root of q (see [1]) It is therefore, natural to extend the ground ring of ~ as follows For any Coxeter group (W, S) we define the Hecke algebra ~ to be J{' | A, where A is the ring of Laurent polynomials with integral coefficients in the indeterminate ql/2 Our purpose is to construct representations oL,Uf endowed with a special basis They will be defined in terms of certain graphs We define a W-graph to be a set of vertices X, with a set Y of edges (an edge is a subset of X consisting of two elements) together with two additional data: for each vertex xeX , we are given a subset I x of S and, for each ordered pair of vertices y, x such that {y, x} e Y, we are given an integer p(y, x) +0 These data are subject to the requirements (10a), (10b) below Let E be

Abstract: 0. Introduction. An n-dimensional convex polytope is simple if the number of codimension-one faces meeting at each vertex is n. In this paper we investigate certain group actions on manifolds, which have a simple convex polytope as orbit space. Let P\" denote such a simple polytope. We have two situations in mind. (1) The group is Z, M\" is n-dimensional and (2) The group is T\", M is 2n-dimensional and M2\"/T\" P\". Up to an automorphism of the group, the action is required to be locally isomorphic to the standard representation of Z. on in the second case. In the first case, we call M a \"small cover\" of P\"; in the second, it is a \"toric manifold\" over P. First examples are provided by the natural actions of Z. and T\" on RP\" and CP\", respectively. In both cases the orbit space is an n-simplex. Associated to a small cover of P\", there is a homomorphism 2\" Z’ Z, where m is the number of codimension-one faces of P\". The homomorphism 2 specifies an isotropy subgroup for each codimension-one face. We call it a \"characteristic function\" of the small cover. Similarly, the characteristic function ofa toric manifold over pn is a map Z --) 7/\". A basic result is that small covers and toric manifolds over P\" are classified by their characteristic functions (see Propositions 1.7 and 1.8). The algebraic topology of these manifolds is very beautiful. The calculation of their homology and cohomology groups is closely related to some well-known constructions in commutative algebra and the combinatorial theory of convex polytopes. We discuss some of these constructions below. Let f denote the number of/-faces of P\" and let h denote the coefficient of \"in f(t 1). Then (fo, f,) is called the f-vector and (ho, h,) the h-vector of P\". The f-vector and the h-vector obviously determine one another. The Upper Bound Theorem, due to McMullen, asserts that the inequality h < (,-,{-x), holds for all n-dimensional convex polytopes with m faces of codimension one. In 1971 McMullen conjectured simple combinatorial conditions on a sequence (h0, h,) of integers necessary and sufficient for it to be the h-vector of a simple convex polytope. The sufficiency of these conditions was proved by Billera and Lee and necessity by Stanley (see [Bronsted] for more details and references). Research on

Abstract: 1 CARTAN MATRICES AND FINITE COXETER GROUPS 2 PARABOLIC SUBGROUPS 3 CONJUGACY CLASSES AND SPECIAL ELEMENTS 4 THE BRAID MONOID AND GOOD ELEMENTS 5 IRREDUCIBLE CHARACTERS OF FINITE COXETER GROUPS 6 PARABOLIC SUBGROUPS AND INDUCED CHARACTERS 7 REPRESENTATION THEORY OF SYMMETRIC ALGEBRAS 8 IWAHORI-HECKE ALGEBRAS 9 CHARACTERS OF IWAHORI-HECKE ALGEBRAS 10 CHARACTER VALUES IN CLASSICAL TYPES 11 COMPUTING CHARACTER VALUES AND GENERIC DEGREES APPENDIX: TABLES FOR THE EXCEPTIONAL TYPES REFERENCES