Root systems, symmetries and linear representations of Artin groups

Abstract

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.