A question about the reflection groups and root system.

Question

I am studing the book "Reflection groups and Coxeter groups" written by James E. Humphreys. On page 11, I am at a loss for the Theorem 1.5.

I don't understand why $W'\beta \cap \Delta\neq \emptyset$ in $(2)$ can explain $\Pi \subset W'\Delta$ ?

I post my effort here. For any $\beta \in \Pi$ $$\beta =s_{\alpha} (s_{\alpha}\beta), $$ we need only to show $s_{\alpha}\beta\subset \Delta$.

Answer 1

\begin{align*} &\, \text{for every $\beta \in \Pi$ : $W' \beta \cap \Delta \neq \emptyset$} \\ \iff&\, \text{for every $\beta \in \Pi$ exists $w' \in W'$ with $w' \beta \in \Delta$} \\ \iff&\, \text{for every $\beta \in \Pi$ exists $w' \in W'$ with $\beta \in (w')^{-1} \Delta$} \\ \iff&\, \text{for every $\beta \in \Pi$ exists $\tilde{w}' \in W'$ with $\beta \in \tilde{w}' \Delta$} \\ \iff&\, \text{for every $\beta \in \Pi$ : $\beta \in W' \Delta$} \\ \iff&\, \Pi \subseteq W' \Delta \end{align*}

A bit less formal: \begin{align*} &\, \text{for every $\beta \in \Pi$ : $W' \beta \cap \Delta$} \\ \iff&\, \text{the $W'$-orbit of every $\beta \in \Pi$ contains some $\alpha \in \Delta$} \\ \iff&\, \text{every $\beta \in \Pi$ is contained in the $W'$-orbit of some $\alpha \in \Delta$} \\ \iff&\, \Pi \subseteq W' \Delta \end{align*}

Answer 2

Having proved $W'\beta\cap \Delta\neq\emptyset$ for any $\beta\in \Pi$ we have the following argument:

Let $\beta\in\Pi$. There exists $w\in W'$ such that $w\beta=\alpha\in\Delta$. Now, $w^{-1}\in W'$ and $\beta=w^{-1}\alpha\in W'\Delta$. It follows that $\Pi\subset W'\Delta$.