Involution products in finite Coxeter groups

Abstract

It is well known that every element $w$ of a finite Coxeter group can be written as a product of at most two involutions. In general it is not possible to choose $x, y$ such that $w = xy$, $x^2 = y^2 = 1$, and $l(w) = l(x) + l(y)$. However one can choose $x, y$ such so that $L(w) = L(x) + L(y)$ (where $L(w)$ is the reflection length of $w$). Given all possible $x, y$ with $x^2 = y^2 = 1$ and $xy = w$, the minimum value of $l(x) + l(y) - l(w)$ is called the excess of $w$; the reflection excess of $w$ is the minimum value over all such $x, y$ for which, additionally, $L(x) + L(y) = L(w)$. The purpose of this article is to investigate the properties of excess and reflection excess. (MSC2000: 20F55)