Zero Excess and Minimal Length in Finite Coxeter Groups

Abstract

Let $\mathcal{W}$ be the set of strongly real elements of $W$, a Coxeter group. Then for $w\in \mathcal{W}$, $e(w)$, the excess of $w$, is defined by $e(w)=\min \{\ell (x)+\ell (y)-\ell (w)|w=xy,x^2=y^2=1\}$. When $W$ is finite we may also define $E(w)$, the reflection excess of $w$. The main result established here is that if $W$ is finite and $X$ is a $W$-conjugacy class, then there exists $w\in X$ such that $w$ has minimal length in $X$ and $e(w)=0=E(w)$.

This is the preprint version of: Zero Excess and Minimal Length in Finite Coxeter Groups, J. Group Theory 15 (2012), no. 4, 497--512