Involution Products in Coxeter Groups

TL;DR

This paper studies involution products in Coxeter groups, showing that every element formed by involutions is conjugate to one with minimal excess, revealing structural properties of these elements.

Contribution

It proves that any involution product in a Coxeter group is conjugate to an element with zero excess, extending known results from finite to infinite groups.

Findings

Every involution product is conjugate to an element with zero excess.

The concept of excess generalizes known properties of finite Coxeter groups.

Structural insights into involution products in Coxeter groups.

Abstract

For $W$ a Coxeter group, let $\mathcal{W} = \{ w \in W \;| \; w = xy \; \mbox{where} \; x, y \in W \; \mbox{and} \; x^2 = 1 = y^2 \}$. If $W$ is finite, then it is well known that $W = \mathcal{W}$. Suppose that $w \in \mathcal{W}$. Then the minimum value of $\ell(x) + \ell(y) - \ell(w)$, where $x, y \in W$ with $w = xy$ and $x^2 = 1 = y^2$, is called the \textit{excess} of $w$ ($\ell$ is the length function of $W$). The main result established here is that $w$ is always $W$-conjugate to an element with excess equal to zero.