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 H{é}e and Mühlherr, $W^G$ 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.