Intrinsic reflections in Coxeter systems

TL;DR

This paper characterizes when right-angled generators in Coxeter systems are intrinsic reflections, providing necessary and sufficient conditions for their identification across all generating sets.

Contribution

It introduces criteria to determine when a right-angled generator is an intrinsic reflection in Coxeter groups, advancing understanding of their structural properties.

Findings

Provides necessary and sufficient conditions for intrinsic reflections.

Clarifies the role of right-angled generators in Coxeter systems.

Enhances classification of Coxeter group symmetries.

Abstract

Let $(W,S)$ be a Coxeter system and let $s \in S$. We call $s$ a right-angled generator of $(W,S)$ if $st = ts$ or $st$ has infinite order for each $t \in S$. We call $s$ an intrinsic reflection of $W$ if $s \in R^W$ for all Coxeter generating sets $R$ of $W$. We give necessary and sufficient conditions for a right-angled generator $s \in S$ of $(W,S)$ to be an intrinsic reflection of $W$.