Root systems, symmetries and linear representations of Artin groups

2018/04/20 by Geneste, Olivier, Hée, Jean-Yves, Paris, Luis
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1804.07519

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 Hée and Mühlherr, W^G 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 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 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.

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