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Coxeter groups and automorphisms
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abstract
Let $(W,S)$ be a Coxeter system and $\Gamma$ be a group of automorphisms of $W$ such that $\gamma(S)=S$ for all $\gamma \in \Gamma$. Then it is known that the group of fixed points $W^\Gamma$ is again a Coxeter group with a canonically defined set of generators. The usual proofs of this fact rely on the reflection representation of $W$. Here, we give a proof which only uses the combinatorics of reduced expressions in $W$. As a by-product, this shows that the length function on $W$ restricts to a weight function on $W^\Gamma$.
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Cited by 1 Pith paper
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Namikawa--Weyl groups of symplectic quotient singularities
Every irreducible Weyl group arises as a factor of the Namikawa–Weyl group of some symplectic quotient singularity V/G with G a symplectic reflection group.
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