Coxeter groups, symmetries, and rooted representations

TL;DR

This paper explores how symmetry groups acting on Coxeter systems induce new Coxeter groups and root bases, providing explicit constructions and showing the faithfulness of the induced representations.

Contribution

It constructs root bases for symmetry-invariant subgroups of Coxeter groups and proves the induced representations are faithful, extending known results with explicit methods.

Findings

W^G is a Coxeter group under symmetry G

Constructed root basis for W^G invariant under G

Induced representation f^G is faithful

Abstract

Let $(W,S)$ be a Coxeter system, let $G$ be a group of symmetries of $(W,S)$ and let $f : W \to \GL (V)$ be the linear representation associated with a root basis $(V, \langle .,. \rangle, \Pi)$.We assume that $G \subset \GL (V)$, and that $G$ leaves invariant $\Pi$ and $\langle .,. \rangle$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde \Pi \subset V^G$ so that $(V^G, \langle .,. \rangle, \tilde \Pi)$ is a root basis of $W^G$, and we show that the induced representation $f^G : W^G \to \GL(V^G)$ is the linear representation associated with $(V^G, \langle .,. \rangle, \tilde \Pi)$.In particular, the latter is faithful. The fact that $W^G$ is a Coxeter group is already known and is due to M\"uhlherr and H\'ee, but also follows directly from the proof of the other results.