A note on involution prefixes in Coxeter groups

TL;DR

This paper investigates the structure of prefixes in Coxeter groups, proving that all Coxeter elements in finitely generated groups have a unique maximal involution prefix, and conjectures this extends to all finite Coxeter group elements.

Contribution

It establishes that Coxeter elements in finitely generated Coxeter groups possess the ancestor property, providing a canonical involution-based expression.

Findings

All Coxeter elements in finitely generated Coxeter groups have the ancestor property.

The set of prefixes of such elements contains a unique involution of maximal length.

Conjecture that the property holds for all non-identity elements of finite Coxeter groups.

Abstract

Let $(W, R)$ be a Coxeter system and let $w \in W$. We say that $u$ is a prefix of $w$ if there is a reduced expression for $u$ that can be extended to one for $w$. That is, $w = uv$ for some $v$ in $W$ such that $\ell(w) = \ell(u) + \ell(v)$. We say that $w$ has the ancestor property if the set of prefixes of $w$ contains a unique involution of maximal length. In this paper we show that all Coxeter elements of finitely generated Coxeter groups have the ancestor property, and hence a canonical expression as a product of involutions. We conjecture that the property in fact holds for all non-identity elements of finite Coxeter groups.