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Coxeter groups, symmetries, and rooted representations

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 $(V, \langle .,. \rangle, Π)$.We assume that $G \subset \GL (V)$, and that $G$ leaves invariant $Π$ and $\langle .,. \rangle$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde Π\subset V^G$ so that $(V^G, \langle .,. \rangle, \tilde Π)$ 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 $(V^G, \langle .,. \rangle, \tilde Π)$.In particular, the latter is faithful. The fact that $W^G$ is a Coxeter group is already known and is due to Mühlherr and Hée, but also follows directly from the proof of the other results.