Root systems, symmetries and linear representations of Artin groups
Résumé
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 Hée and Mühlherr,WG is a Coxeter group associated to some Coxeter graph Γ. We denote by Φ+ the set of positive roots of Γ and by Φ+ the set of positive roots of bΓ. 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 EG contains a linearly independent family of vectors naturally in one-to-one correspondence with Φ+ and we determine exactly when this family is a basis of EG. This question is motivated by the construction of Krammer’s style linear representations for non simply laced Artin groups. © Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal, 2019, Certains droits réservés.