# Coxeter groups, symmetries, and rooted representations

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 $(V, \langle .,. \rangle, \Pi)$. We assume that $G \subset \GL (V)$, and that $G$ leaves invariant $\Pi$ and $\langle .,. \rangle$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde \Pi \subset V^G$ so that $(V^G, \langle .,. \rangle, \tilde \Pi)$ 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 $(V^G, \langle .,. \rangle, \tilde \Pi)$. 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.
Type de document :
Pré-publication, Document de travail
2016

https://hal.archives-ouvertes.fr/hal-01404050
Contributeur : Luis Paris <>
Soumis le : lundi 28 novembre 2016 - 11:53:58
Dernière modification le : mardi 29 novembre 2016 - 01:01:22
Document(s) archivé(s) le : lundi 27 mars 2017 - 09:12:47

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GenParCoxV3.pdf
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### Identifiants

• HAL Id : hal-01404050, version 1
• ARXIV : 1611.09150

### Citation

Olivier Geneste, Luis Paris. Coxeter groups, symmetries, and rooted representations. 2016. <hal-01404050>

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