# Monday October 7

Last time: ?

**Lemma 10.2**

a. ?
b. $\alpha \in \Pi \implies S_\alpha \actson \Phi^+\setminus\theset{\alpha}$ by permutation
c. $\alpha_i \in \Pi$ and $S_{\alpha_1}, \cdots, S_{\alpha_{j-1}}(\alpha_j) \in \Phi^-$ then 
$S_{\alpha_1} \cdots S_{\alpha_j} = S_{\alpha_1} \cdots S_{alpha_{t-1}} \cdots S_{\alpha_{j-1}}$ 
for some $t$, where the former has $j$ terms and the latter has $j-2$ terms.

*Proof of (a): ?*

*Proof of (b):*

Suppose towards a contradiction that $w(\alpha_j) \in \Phi^+$.
Then consider $WS(\alpha_j) = -W(\alpha_j) \in \Phi^-$.

By Lemma 10.2(c), we have
$W = S_{\alpha_1} \cdots S_{alpha_{t-1}} S_{\alpha_{t+1}} \cdots S_{\alpha_{j-1}} S_{\alpha_j}$,
where this is $j-1$ terms.
So $w = S_\alpha \cdots S_{\alpha_j}$ is not reduced.

## Weyl Groups

Recall that the *chambers* are given by the connected component of $E \setminus \union_{\alpha\in\Phi} H_\alpha$.

**Theorem:**
Fix $\Pi$ of $\Phi$. 
Then

a. $W \actson\theset{\text{chambers}}$ transitively
b. $W \actson\theset{\text{bases}}$ transitively
c. $\forall \alpha \Phi, ~\exists w\in W \suchthat w(\alpha) \in Pi$
d. $W \coloneqq \theset{S_\alpha \suchthat \alpha\in\Phi} = \generators{S_\alpha \mid \alpha \in \Pi} \coloneqq W_0$
e. $W \actson \theset{\text{bases}}$ simply transitively, i.e. $w(\Pi) = \Pi \implies w = e$.

> I.e. we can describe the Weyl group using only simple reflections

*Proof:*
We will prove (a) -- (c)$ for $W_0$.

*Proof of (a):*
Recall the fundamental chamber,
$C(\Pi) = \theset{x\in E \suchthat (x, a) > 0 ~\forall \alpha\in\Pi}$.
We want to show that any chamber $C$ is equal to $wC(\Pi)$.

Pick $\gamma \in C$ and $g\in W_0$ such that 
$(g(\gamma), \rho) = \max\theset{(w(\gamma), \rho) \suchthat w\in W_0}$,
which exists because $W_0$ is a finite group.

For all $\alpha \in \Pi, S_\alpha g \in W_0$ and so by maximality we have
\begin{align*}
(g(\gamma), \rho) 
\geq (s_\alpha g(\gamma) \rho) \\
= (g(\gamma), S_\alpha(\rho)) \\
= (g(\gamma), \rho - \alpha) \\
= (g(\gamma), \rho) - (g(\gamma), \alpha)
.\end{align*}

and so $(g(\gamma), 0) \geq 0$, because this can never be an equality since $\gamma \in C$.
Thus $g(\gamma) \in C(\Pi)$.

*Proof of (b):*

This holds because there a correspondence between $\theset{C(\Pi)} \iff \theset{\text{bases} \Pi}$.

*Proof of (c):*

It suffices to show that $\alpha \in \Phi$ lies in some base $\Pi' = W(\Pi)$.
Note that $\beta\neq\alpha \implies H_\beta \neq H_\alpha$, and so we can pick a $\gamma \in H_\alpha \intersect H_\beta^c$ for every $\beta \in \Phi \setminus{\pm \alpha}$.
Since $\inner{\gamma}{\alpha} = 0$ but $\inner{\gamma}{\beta} \neq 0$ for all $\beta \neq \pm \alpha$, we can choose some $\varepsilon > 0$ such that $\abs{\inner{\gamma'}{\beta}} > \varepsilon$ for every $\beta\neq\pm\alpha$.
Then $\gamma' \in C(\Pi')$ and thus $\alpha \in \Pi'$.

*Proof of (d):*

By definition, $W_0 \subseteq W$, so we need to show the reverse containment.
For all $\alpha \in Phi$, we want to show $S_\alpha \in W_0$.
By (c), there exists a $w\in W_0$ such that $w(\alpha) \coloneqq \beta \in \Pi$,
Then $S_\beta = S_{w(\alpha)} = w s_\alpha w\inv$.
So $S_\alpha = w\inv S_\beta w$, where each term is in $W_0$, so the whole thing is in $W_0$ as well.

*Proof of (e):*

Suppose $W(\Pi) = \Pi$.
Let $W = S_{\alpha_1} \cdots S_{\alpha_\ell}$ be a reduced expression, which exists by (d).
By corollary 10.2b, we have $W(\alpha_\ell \in \Phi^-)$.
But this forces $w = e$. 

$\qed$

*Remarks:*

By (d), there is a well-defined notion of *length* for $w\in W$.
We will now show that $\ell(w) = n(w) \coloneqq \# N_w \coloneqq \# \theset{\alpha \in \Phi^+ \suchthat W(\alpha) \in \Phi^-}$, i.e. the number of roots that get sent to a negative root.