# A word property for twisted involutions in Coxeter groups

## TL;DR

This paper introduces a minimal set of moves to connect all reduced expressions of twisted involutions in Coxeter groups, extending the classical word property and improving previous results.

## Contribution

It provides a new minimal move set for twisted involutions in Coxeter groups, generalizing and improving existing results on the word property.

## Key findings

- Connected all reduced involution words using minimal moves
- Extended classical Coxeter word property to twisted involutions
- Generalized previous results in specific Coxeter types

## Abstract

Given an involutive automorphism $\theta$ of a Coxeter system $(W,S)$, let $\mathfrak{I}(\theta) \subseteq W$ denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced $\underline{S}$-expressions (also known as admissible sequences, reduced $I_\theta$-expressions, or involution words) for any given $w \in \mathfrak{I}(\theta)$. This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg.