arXiv subjects

Olivier Geneste

Publications and source records attributed to Olivier Geneste.

2 recordsLinked to original sources

arXiv2018

Root systems, symmetries and linear representations of Artin groups

Let $Γ$ be a Coxeter graph, let $W$ be its associated Coxeter group, and let $G$ be a group of symmetries of $Γ$.Recall that, by a theorem of H{é}e and Mühlherr, $W^G$ is a Coxeter group associated to some Coxeter graph $\hat Γ$.We denote by $Φ^+$ the set of positive roots of $Γ$ and by $\hat Φ^+$ the set of positive roots of $\hat Γ$.Let $E$ be a vector space over a field $\K$ having a basis in one-to-one correspondence with $Φ^+$.The action of $G$ on $Γ$ induces an action of $G$ on $Φ^+$, and therefore on $E$.We show that $E^G$ contains a linearly independent family of vectors naturally in one-to-one correspondence with $\hat Φ^+$ 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.

math.GR↗

arXiv2016

Coxeter groups, symmetries, and rooted representations

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 $(V, \langle .,. \rangle, Π)$.We assume that $G \subset \GL (V)$, and that $G$ leaves invariant $Π$ and $\langle .,. \rangle$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde Π\subset V^G$ so that $(V^G, \langle .,. \rangle, \tilde Π)$ 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 $(V^G, \langle .,. \rangle, \tilde Π)$.In particular, the latter is faithful. The fact that $W^G$ is a Coxeter group is already known and is due to Mühlherr and Hée, but also follows directly from the proof of the other results.

math.GR↗