A note on involution prefixes in Coxeter groups
Sarah B. Hart, Peter J. Rowley
TL;DR
The paper investigates when prefixes of an element $w$ in a Coxeter group yield a unique involution prefix, introducing the ancestor property via $A(w)$ and the involution length $\ell_{\text{inv}}(w)$. It defines involution prefixes $\mathcal{IP}(w)$ and shows, for finitely generated Coxeter groups, that every Coxeter element has a unique ancestor, given by $a(w)=\prod_{r\in D(w)} r$ where $D(w)$ is the left descent set. This leads to a precise connection between $\ell_{\text{inv}}(w)$ and the Coxeter graph: $\ell_{\text{inv}}(w)$ equals the path length of a maximal $w$-path, with the minimum equal to the chromatic number $\chi(\Gamma)$ and the maximum to the graph’s maximum path length; certain finite types attain the rank as the maximum. The authors verify the conjectures for many finite irreducible groups via computational evidence and discuss extensions to reducible finite groups, while providing counterexamples in infinite Coxeter groups, illustrating a stark contrast between finite and infinite cases. Overall, the work yields a canonical involution-based decomposition for Coxeter elements and links algebraic properties to graph-theoretic invariants, informing both Bruhat-order structure and involution-length behavior.
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.