# Root systems, symmetries and linear representations of Artin groups

Abstract : Let $\Gamma$ be a Coxeter graph, let $W$ be its associated Coxeter group, and let $G$ be a group of symmetries of $\Gamma$. Recall that, by a theorem of Hée and M\"uhlherr, $W^G$ is a Coxeter group associated to some Coxeter graph $\hat \Gamma$. We denote by $\Phi^+$ the set of positive roots of $\Gamma$ and by $\hat \Phi^+$ the set of positive roots of $\hat \Gamma$. Let $E$ be a vector space over a field $\K$ having a basis in one-to-one correspondence with $\Phi^+$. The action of $G$ on $\Gamma$ induces an action of $G$ on $\Phi^+$, and therefore on $E$. We show that $E^G$ contains a linearly independent family of vectors naturally in one-to-one correspondence with $\hat \Phi^+$ and we determine exactly when this family is a basis of $E^G$. This question is motivated by the construction of Krammer's style linear representations for non simply laced Artin groups.
Type de document :
Pré-publication, Document de travail
2018

https://hal.archives-ouvertes.fr/hal-01770502
Contributeur : Luis Paris <>
Soumis le : jeudi 19 avril 2018 - 09:56:55
Dernière modification le : vendredi 8 juin 2018 - 14:50:07
Document(s) archivé(s) le : mardi 18 septembre 2018 - 11:37:51

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180418GeHePaV3.pdf
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### Identifiants

• HAL Id : hal-01770502, version 1
• ARXIV : 1804.07519

### Citation

Olivier Geneste, Jean-Yves Hée, Luis Paris. Root systems, symmetries and linear representations of Artin groups. 2018. 〈hal-01770502〉

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