Root systems, symmetries and linear representations of Artin groups

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