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.