A note on involution prefixes in Coxeter groups

2025/02/02 by Sarah B. Hart, Hart, Sarah B., Peter Rowley +1
Computer Science · Mathematics · #20F55 #Advanced Combinatorial Mathematics #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2502.00777

openalex publication_date 2025/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (W, R) be a Coxeter system and let w ∈ 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 ℓ(w) = ℓ(u) + ℓ(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.

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