Root systems, symmetries and linear representations of Artin groups

TL;DR

This paper explores the relationship between root systems, symmetries, and linear representations of Artin groups, establishing a connection between invariant subspaces and positive roots in Coxeter groups, with implications for constructing linear representations.

Contribution

It provides a detailed analysis of the invariant subspace structure of root systems under group actions, linking it to Coxeter group symmetries and aiding in the construction of linear representations for Artin groups.

Findings

Identifies a linearly independent family in the invariant subspace corresponding to positive roots.

Determines conditions under which this family forms a basis of the invariant subspace.

Connects the structure of invariant subspaces to the construction of Krammer's style linear representations.

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{\'e}e and M\"uhlherr, $W^G$ 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.