# Wednesday October 2

**Recall from last time:**

1. $\abs \Phi < \infty$ and $\Phi$ spans $E$, where $0\not\in \Phi$
2. If $\alpha \in \Phi$, then $C\alpha \intersect \Phi = \theset{\pm \alpha}$
3. $\alpha \in \Phi$, then $S_\alpha(\Phi) = \Phi$.
3. If $\alpha, \beta \in \Phi$, then $\inner{\beta}{\alpha\dual} \in \ZZ$
  where $(E, \inner{\wait}{\wait})$is Euclidean and
\begin{align*}
S_\alpha: E \to E \\
\beta \mapsto \beta - \inner{\beta}{\alpha\dual}\alpha, \quad \alpha\dual = \frac {2} {\inner{\alpha}{\alpha}} \alpha
.\end{align*}

*Examples:*

In Rank 1:

1. Prop 2 implies $\Phi = \theset{\pm \alpha}$
2. Prop 1 implies $E = \RR \alpha$
3. Prop 3: $S_\alpha(\alpha) = -\alpha$
4. Prop 4 implies $\inner{\pm\alpha}{\pm\alpha} = \pm \frac {2\inner{\alpha}{\alpha}} {\inner{\alpha}{\alpha}} = \pm 2$

*Rank 1 Diagram:*

> Todo: Insert phone image


*In Rank 2:*

> Todo: Insert phone image


*Exercise:*

- Show that $\mathrm{ord}(S_\alpha, S_\beta) = 2,3,4,6$ for types $A_1 \cross A_1, B_2, G_2$.
- Show that $W(A_2) \cong \ZZ_3$ and $W(B_2) \cong D_8$.

## Pairs of Roots

**Lemma:**
Let $\alpha, \beta \in \Phi$ where $\beta \neq \pm \alpha$, then 

1. $\inner{\alpha}{\beta\dual} \inner{\beta}{\alpha\dual} \in \theset{0,1,2,3}$
  Moreover, assuming $\abs \beta \geq \abs \alpha$, we have the following table
  
  > Todo: Insert table

2. If $\inner{\alpha}{\beta} > 0$, then $\alpha-\beta \in\Phi$. Similarly, if $\inner{\alpha}{\beta} > 0$, then $\alpha + \beta \in \Phi$.

3. Any root string is unbroken and has length greater than 4.

*Proof of (1):*

By the Law of Cosines, we can write $x \coloneqq \inner{\beta}{\alpha\dual} \inner{\alpha}{\beta\dual} = 4\cos^2(\theta) \in \ZZ$.
This restricts the possibilities to $x \leq 4$. 
But $x = 4 \iff \alpha = c \beta$, i.e. $\theta = 0$, but we are assuming that $\alpha \neq \pm \beta$, so this can not happen.

*Proof of (2):*

Since $\inner{\alpha}{\beta} > 0$ and $\abs \beta \geq \abs \alpha$, then $\inner{\alpha}{\beta\dual} = 1$.
But then $S_\beta(\alpha) = \alpha - \inner{\alpha, \beta\dual}\beta \in\Phi$ by Prop 3.
So this is equal to $\alpha - \beta$.

A similar argument works for $\abs \beta \leq \abs \alpha$.

*Proof of (3):*
Let $p,q$ be the largest integers such that $b-p\alpha, b+q\alpha \in \Phi$ respectively.
Suppose that the root stream between these two is broken somewhere, say $\beta + s\alpha \in \Phi$ and $\beta + (s+1)\alpha \not\in\Phi$ by counting up from $\beta - p\alpha$.
Similarly, there is some $t$ counting down from $b+q\alpha$ then $\beta + t\alpha \in \Phi$ but $\beta + (t-1)\alpha \not\in\Phi$.
In particular, $s < t$.
From (2), we have 
$\inner{\alpha}{\beta + s\alpha} \geq 0$,
$\inner{\alpha}{\beta + t\alpha} \leq 0$.

We have
$$
\inner{\alpha}{\beta} + t\inner{\alpha}{\alpha} = \inner{\alpha}{\beta + t\alpha} \leq 0 \leq \inner{\alpha}{beta + s\alpha} = \inner{\alpha}{\beta} + s\inner{\alpha}{\alpha}
$$

where we know that $\inner{\alpha}{\alpha} > 0$.

Since $S_\alpha(\Phi) = \Phi$ and these $S_\alpha(\beta + i\alpha) = \beta - \ZZ \alpha$, we find that reflections permute the root string.
We then find that $p = \inner{\beta}{\alpha\dual} + q$, and so $\inner{\beta}{\alpha\dual} = p-q \in [-3, 3]$.

## Chapter 10: Simple Roots and Weyl Groups

**Definition:**
A *base* of a root system $\Phi$ is a subset $\Pi \subseteq \Phi$ such that 

1. $\Pi$ is a basis for the underlying vector space $E$, and
2. Each $\beta \in \Phi$ can be written as $\beta = \sum_{\alpha\in\Pi} \kappa_\alpha^\beta \alpha$ where all of the coefficients $\kappa_\alpha^\beta$ all have the same sign.

The roots in $\Pi$ are called *simple*.
A root $\beta$ is *positive* (resp. *negative*) if the $\kappa_\alpha^\beta \geq 0$ for all $\beta in \Phi^+$ (resp $\leq 0$ in $\Phi^-$).
The *height* of a $\beta$ is the sum of the coefficients.
$\Pi$ defines a partial order on $E$ where $\mu \leq \lambda \iff \lambda - \mu \in \sum_{\alpha \in \Pi} \ZZ_{\geq 0} \alpha$.

> Note that this is defined on the roots themselves, and can then be extended to all of $E$.

> Todo: Insert phone image