I came across two statements while studying Weyl group of root Systems.
First one: The Weyl group say $W$ sends one Weyl Chamber onto another. If $\gamma$ is regular in a Euclidean Space $E$, we have $\sigma(C(\gamma))=C(\sigma \gamma))$, if $\sigma \in W$. Any idea for how to prove this identity and see it geometrically?
Second one: $W$ sends a base $∆$ to $\sigma(∆)$ which is again a base. Although somehow it follows from the identity $\sigma(∆(\gamma))=∆(\sigma\gamma)$ which follows from $(\sigma \gamma, \sigma \alpha)=(\gamma, \alpha)$(How?).
I don't have any clear idea of these two points. It will be great if someone can help me in these!