# Monday, October 10

## Classification of Rank 2 Root Systems

:::{.remark}
If $\beta\neq \pm \alpha$,

- \( (\alpha, \beta) > 0 \implies \alpha - \beta\in \Phi \) 
- \( (\alpha, \beta) < 0 \implies \alpha + \beta\in \Phi \) 

:::

:::{.remark}
Rank 2 root systems: let $\alpha$ be a root of shortest length, and $\beta$ a root with angle $\theta$ between $\alpha,\beta$ with $\theta \geq \pi/2$ as large as possible.

- If $\theta = \pi/2$: $A_1 \times A_1$.

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  ![](figures/2022-10-10_09-16-04.png)

- If $\theta = 2\pi/3$: $A_2$

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![](figures/2022-10-10_09-20-14.png)

  One can check $\inp \alpha \beta= 2(-1/2) = -1$ and $\inp {\alpha + \beta}{ \beta} = \inp \alpha \beta + \inp \beta \beta = -1 + 2 = 1$.

- If $\theta = 3\pi/4$: $B_2$

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- If $\theta = 5\pi/6$: $G_2$

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  One can check that using linearity of $\inp\wait\wait$ in the first variable that

  - $s_\alpha \beta = \beta + 3 \alpha$, 
  - $s_\alpha(\beta+ \alpha) = \beta+ 2 \alpha$,
  - $s_ \beta(\beta+ 3 \alpha) = (\beta+ 3 \alpha) - \inp{ \beta+ 3 \alpha}{ \beta}= 2 \beta+ 3 \alpha \in \Phi$.

:::

:::{.remark}
Note that in each case one can see the root strings, defined as 
\[
R_\beta \da \ts{\beta+ k \alpha \st k\in \ZZ} \intersect \Phi
.\]
Let $r,q\in \ZZ_{\geq 0}$ be maximal such that $\beta-r \alpha, \beta + q \alpha\in \Phi$.
The claim is that every such root string is unbroken.

Suppose not, then there is some $k$ with $-r < k < q$ with $\beta + k \alpha \not\in \Phi$.
One can then find a maximal $p$ and minimal $s$ with $p < s$ and $\beta+p \alpha \in \Phi$ but $\beta + (p+1) \alpha \not\in \Phi$, and similarly $\beta + (s-1)\alpha\not\Phi$ but $\beta + s \alpha\in \Phi$.
By a previous lemma, $(\beta+ p \alpha, \alpha) \geq 0$ and similarly $(\beta+ s \alpha, \alpha) \leq 0$.
Combining these, 
\[
p( \alpha, \alpha) \geq s (\alpha, \alpha) \implies p \geq s \text{ since } (\alpha, \alpha) > 0 \qquad\contradiction
.\]

The picture:

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So $s_\alpha$ reverses the root string, since it sends the line containing the root string to itself but reflects through $P_\alpha$.
One can compute
\[
\beta - r \alpha 
&= s_ \alpha(\beta + q \alpha) \\
&= (\beta+ q \alpha) - \inp{\beta+ q \alpha}{ \alpha} \alpha \\
&= (\beta+ q \alpha) - \inp{ \beta}{\alpha}ga - 2q \alpha \\
&= \beta - \qty{\inp \beta \alpha + q} \alpha
,\]
so $r = \inp \beta \alpha$ and $r-q = \inp \beta \alpha = \beta(h_ \alpha)$ for a semisimple Lie algebra.

Supposing \( \abs{\inp \beta \alpha} \leq 3 \).
Choose $\beta$ in $R_\alpha$ such that $\beta-\alpha$ is not a root and $\beta$ is at the left end of the string and $r=0$:

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![](figures/2022-10-10_09-51-39.png)

Then $q = -\inp \beta \alpha$, so the root string contains at most 4 roots (for $\Phi$ of any rank).
:::

## Ch. 10 Simple Roots, 10.1 Bases and Weyl Chambers

:::{.definition title="Base (simple roots)"}
A subset $\Delta \subseteq \Phi$ is a **base** (or more modernly a **set of simple roots**) if 

- B1: $\Delta$ is a basis for $\EE$,
- B2: Each $\beta\in \Phi$ can be written as $\beta = \sum_{\alpha\in \Delta} k_\alpha \alpha$ with $k_\alpha\in \ZZ$ with either all $k_\alpha\in \ZZ_{\geq 0}$ or all $k_\alpha \in \ZZ_{\leq 0}$.
:::

:::{.example title="?"}
1. The roots labeled $\alpha,\beta$ in the rank 2 cases were all simple systems. 
2. For $A_n$, a base is $\ts{\eps_1 - \eps_2, \eps_2 - \eps_3, \cdots, \eps_{n} - \eps_{n+1} }$, where $\Phi = \ts{\eps_i - \eps_j \st i\neq j}$.

:::