# Coxeter groups, symmetries, and rooted representations

1 IMB - Institut de Mathématiques de Bourgogne Dijon

Abstract : Let $W,S$ be a Coxeter system, let $G$ be a group of symmetries of $W,S$ and let $f : W \to \GL V$ be the linear representation associated with a root basis $V, \langle .,. angle, \Pi$.We assume that $G \subset \GL V$, and that $G$ leaves invariant $\Pi$ and $\langle .,. angle$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde \Pi \subset V^G$ so that $V^G, \langle .,. angle, \tilde \Pi$ is a root basis of $W^G$, and we show that the induced representation $f^G : W^G \to \GL V^G$ is the linear representation associated with $V^G, \langle .,. angle, \tilde \Pi$.In particular, the latter is faithful. The fact that $W^G$ is a Coxeter group is already known and is due to M\-uhlherr and H\-ee, but also follows directly from the proof of the other results.

Author: Olivier Geneste - Luis Paris -

Source: https://hal.archives-ouvertes.fr/