Maximal length elements of excess zero in finite Coxeter groups

TL;DR

This paper proves that in finite Coxeter groups, each conjugacy class contains elements of both minimal and maximal length with zero excess, extending previous results about minimal length elements.

Contribution

It establishes the existence of maximal length elements with zero excess in every conjugacy class of finite Coxeter groups, complementing earlier findings on minimal length elements.

Findings

Every conjugacy class has a maximal length element with zero excess.

The concept of excess extends to strongly real classes in infinite Coxeter groups.

The results unify the understanding of element lengths within conjugacy classes.

Abstract

The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) - l(w)$, where $x$, $y$ are elements of $W$ such that $x^2 = y^2 = 1$ and $w = xy$. Every element of a finite Coxeter group is either an involution or the product of two involutions, so the concept of excess is well defined. It can be extended to strongly real classes of infinite Coxeter groups. Earlier work by the authors showed that every conjugacy class of a finite Coxeter group contains an element of minimal length and excess zero. The current paper shows that each conjugacy class also contains an element of maximal length and excess zero.