A note on involution prefixes in 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.