On Excess in Finite Coxeter Groups

Abstract

For a finite Coxeter group $W$ and $w$ an element of $W$ the `excess' of $w$ is defined to be $e(w)=\min \{\ell (x)+\ell (y)-\ell (w)|w=xy,x^2=y^2=1\}$ where $\ell$ is the length function on $W$. Here we investigate the behaviour of $e(w)$, and a related concept reflection excess, when restricted to standard parabolic subgroups of $W$. Also the set of involutions inverting $w$ is studied.

This is a preprint version. It has been accepted, subject to revision, in J. Pure Appl. Alg