The affine Weyl group ( e Cn, S) can be realized as the fixed point set of the affine Weyl group ( e A2n−1, e S) under a certain group automorphism α with α( S) = e S. Let e ` be the length function of e A2n−1. We study the cells of the weighted Coxeter group ( e Cn, e `). The main results of the paper are to give an explicit description for all the cells of ( e Cn, e `) corresponding to the pa...

Reflection Groups The theory of abstract reflection groups is due to Tits [281]. What is the appropriate notion of an “abstract reflection group”? At first approximation, one might consider pairs (W, S), where W is a group and S is any set of involutions which generates W. This is obviously too broad a notion. Nevertheless, it is a step in the right direction. In Chapter 3, we shall call such a...

Extended Abstract 1 Preliminaries. Following the spectral graph theory, a graph coloring technique and algebraic methods in graph theory (see [5], [13]), we continue a Coxeter spectral study the category Bigrn of connected non-negative loop-free edge-bipartite (signed) graphs ∆, with n ≥ 1 vertices (bigraphs, in short), and their morsifications introduced by the second named author in [22]-[25]...

Groups defined by presentations of the form 〈s1, . . . , sn | si = 1, (sisj)i,j = 1 (i, j = 1, . . . , n)〉 are called Coxeter groups. The exponents mi,j ∈ N ∪ {∞} form the Coxeter matrix, which characterizes the group up to isomorphism. The Coxeter groups that are most important for applications are the Weyl groups and affine Weyl groups. For example, the symmetric group Sn is isomorphic to the...

A Coxeter group is rigid if it cannot be defined by two nonisomorphic diagrams. There have been a number of recent results showing that various classes of Coxeter groups are rigid, and a particularly interesting example of a nonrigid Coxeter group has been given in [17]. We show that this example belongs to a general operation of “diagram twisting”. We show that the Coxeter groups defined by tw...

We consider Lusztig’s a-function on Coxeter groups (in the equal parameter case) and classify all Coxeter groups with finitely many elements of a-value 2 in terms of Coxeter diagrams.

At the conference the author gives a talk which surveys the definition, history and preceding results on the isomorphism problem of Coxeter groups (problem of deciding which Coxeter groups are isomorphic), together with backgrounds for Coxeter groups, applications of this problem and some related topics in the theory of Coxeter groups, including the author’s recent works. The author would like ...

Let (W,S) be a Coxeter system and let w → w∗ be an involution of W which preserves the set of simple generators S. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements w ∈ W with w−1 = w∗) naturally generates a module of the Hecke algebra of (W,S) with two distinguished bases. The transition matrix between these bases defines a family of polynomials Pσ y,w w...

We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the ‘usual’ nil-Coxeter algebras: a novel 2-parameter type A family that we call NCA(n, d). We explore several combinatorial properties of NCA(n, d), including its Coxeter word basis, length...

Let (W,R) be an arbitrary Coxeter system. We determine the number of elements of W that have a unique reduced expression.