Given a Coxeter system (W,S) equipped with an involutive automorphism θ, the set of twisted identities is ι(θ) = {θ(w)w | w ∈ W}. We point out how ι(θ) shows up in several contexts and prove that if there is no s ∈ S such that sθ(s) is of odd order greater than 1, then the Bruhat order on ι(θ) is a graded poset with rank function ρ given by halving the Coxeter length. Under the same condition, ...

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge in the finite case. In this work we deal with any finite or affine Coxeter group W , and we enumerate fully commutative elements according to their Coxeter length. Our appro...

We give a quadratic lower bound and a cubic upper bound on the order dimension of the Bruhat (or strong) ordering of the affine Coxeter group Ãn. We also demonstrate that the order dimension of the Bruhat order is infinite for a large class of Coxeter groups.

We give a conceptual proof that the Poincaré series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using triangulated categories and spherical twist functors.

A recent form of the Todd-Coxeter algorithm, known as the lookahead algorithm, is described. The time and space requirements for this algorithm are shown experimentally to be usually either equivalent or superior to the Felsch and HaselgroveLeech-Trotter algorithms. Some findings from an experimental study of the behaviour of Todd-Coxeter programs in a variety of situations are given.

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig’s G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [GKP].

We obtain the equivariant K-homology of the classifying space EW for W a right-angled or, more generally, an even Coxeter group. The key result is a formula for the relative Bredon homology of EW in terms of Coxeter cells. Our calculations amount to the K-theory of the reduced C∗-algebra of W , via the Baum-Connes assembly map.

An efficient and purely combinatorial algorithm for calculating products in arbitrary Coxeter groups is presented, which combines ideas of Fokko du Cloux and myself. Proofs are largely based on geometry. The algorithm has been implemented in practical Java programs, and runs surprisingly quickly. It seems to be good enough in many interesting cases to build the minimal root reflection table of ...