## 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.

: Primary 20F55.

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 Wednesday, November 23, 2016