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Consider $\Phi$ a root system (definition of Erdmann) with basis $\Delta \subseteq \Phi$ (subset which is a basis of $E$ and each element of $\Phi$ is a linear combination with only non negative coefficients or non positive coefficients) in an Euclidean space $E$ and $w \in \operatorname{W}(\Phi)$ such that $w(\Delta) = \Delta$.

How can I show that then $w = \mathrm{id}$? I know that the Weyl group is generated by reflections $\sigma_{\alpha}$ with $\alpha \in \Delta$ so we can write $w$ as a composition of reflections, how can I use this to my advantage?

Jendrik Stelzner
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raisinsec
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    I take it that you do not yet have the theorem that the Weyl group acts simply transitively on the bases? (Probably we are trying to prove part of that right here.) – Torsten Schoeneberg Dec 07 '22 at 18:54
  • Yes in my lecture notes we use what the post is about to show that the action is simply transitive but this part is left as an exercise. – raisinsec Dec 07 '22 at 19:03

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