Coxeter groups and automorphisms

Understand

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 $W$.