Intrinsic reflections and strongly rigid Coxeter systems

R. B. Howlett, B. Mühlherr and K. Nuida

Abstract

It is possible for a group $W$ that is abstractly isomorphic to a Coxeter group to have more than one conjugacy class of Coxeter generating sets, and if $S$ and $R$ are two non-conjugate Coxeter generating sets then it may or may not be the case that some element $s\in S$ is conjugate to an element $r\in R$. In this paper we classify the so-called intrinsic reflections: those elements of $W$ whose conjugacy class intersects nontrivially every Coxeter generating set. In combination with previously known results, this leads us to a classification of of Coxeter groups for which all Coxeter generating sets are conjugate.

Keywords: Coxeter group, rigid Coxeter system.

AMS Subject Classification: Primary 20F55.