On Excess in Finite Coxeter Groups

TL;DR

This paper investigates the concept of excess in finite Coxeter groups, analyzing its behavior within parabolic subgroups and exploring related involution properties to deepen understanding of the group's structure.

Contribution

It introduces a detailed study of excess and reflection excess in finite Coxeter groups, focusing on their behavior in parabolic subgroups and involution sets, which is a novel analysis.

Findings

Behavior of excess and reflection excess within parabolic subgroups elucidated.

Characterization of involutions inverting elements in Coxeter groups provided.

Insights into the structure of finite Coxeter groups through excess analysis obtained.

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.