Interval groups related to finite Coxeter groups I

Essays on Coxeter groups Coxeter elements in finite Coxeter groups

A finite Coxeter group possesses a distinguished conjugacy class of Coxeter elements. The literature about these is very large, but it seems to me that there is still room for a better motivated account than what exists. The standard references on thismaterial are [Bourbaki:1968] and [Humphreys:1990], butmy treatment follows [Steinberg:1959] and [Steinberg:1985], from which the clever parts of ...

Character Sums Associated to Finite Coxeter Groups

The main result of this paper is a character sum identity for Coxeter arrangements over finite fields which is an analogue of Macdonald’s conjecture [19] proved by Opdam [21]. 0. Introduction Throughout this paper F will denote a finite field of characteristic p different from 2. We first slightly reformulate Macdonald’s conjecture in a form which has a direct analogue over F. Let G be a finite...

Shortest path poset of finite Coxeter groups

We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group W from the identity to the longest word in W , w0. We show that this poset is the union of Boolean posets of rank absolute length of w0; that is, any shortest path labeled by reflections t1, . . . , tm is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in ...

Automorphisms of nearly finite Coxeter groups

Suppose that W is an infinite Coxeter group of finite rank n, and suppose that W has a finite parabolic subgroup WJ of rank n 1. Suppose also that the Coxeter diagram of W has no edges with infinite labels. Then any automorphism of W that preserves reflections lies in the subgroup of AutðWÞ generated by the inner automorphisms and the automorphisms induced by symmetries of the Coxeter graph. If...

Deeating Innnite Coxeter Groups to Nite Groups