Zero Excess and Minimal Length in Finite Coxeter Groups

TL;DR

This paper investigates the properties of strongly real elements in finite Coxeter groups, demonstrating the existence of elements with minimal length and zero excess and reflection excess within each conjugacy class.

Contribution

It proves that in finite Coxeter groups, each conjugacy class contains an element of minimal length with zero excess and reflection excess, advancing understanding of element structure.

Findings

Existence of minimal length elements with zero excess in each conjugacy class

Existence of minimal length elements with zero reflection excess in each conjugacy class

Characterization of strongly real elements 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 $e(w) = 0 = E(w)$.