# Wednesday, October 05

## Reflections in $\EE^n$

:::{.remark}
Recall the formula
\[
s_\alpha( \lambda) = \lambda- (\lambda, \alpha\dual)\alpha, \qquad \alpha\dual \da {2\alpha\over (\alpha, \alpha)}, \alpha\neq 0
,\]
 which is a reflection through the hyperplane $P_\alpha\da \alpha\perp$:

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\end{tikzpicture}


:::

:::{.lemma title="?"}
Let \( \Phi \subseteq \EE \) be a set that spans $\EE$, and suppose all of the reflections $s_\alpha$ for \( \alpha \in \Phi \) leave $\Phi$ invariant.
If \( \sigma\in \GL(\EE) \) leaves $\Phi$ invariant, fixes a hyperplane $P$ pointwise, and sends some $\alpha\in \Phi\smz$ to $-\alpha$, then $\sigma = s_\alpha$ and $P = P_\alpha$.
:::

:::{.proof title="?"}
Let $\tau = \sigma s_ \alpha =\sigma s_{ \alpha}\inv \in \GL(\EE)$, noting that every $s_\alpha$ is order 2.
Then $\tau( \Phi) = \Phi$ and $\tau( \alpha) = \alpha$, so $\tau$ acts as the identity on the subspace $\RR\alpha$ and the quotient space $\EE/\RR\alpha$ since there are two decompositions $\EE = P_ \alpha \oplus \RR\alpha = P \oplus \RR \alpha$ using $s_\alpha$ and $\sigma$ respectively.
So $\tau - \id$ acts as zero on $\EE/\RR\alpha$, and so maps $\EE$ into $\RR\alpha$ and $\RR\alpha$ to zero, s $(\tau - \id)^2 = 0$ on $\EE$ and its minimal polynomial $m_\tau(t)$ divides $f(t) \da (t-1)^2$.

Note that $\Phi$ is finite, so the vectors \( \beta, \tau \beta, \tau^2 \beta, \tau^3 \beta,\cdots \) can not all be distinct.
Since $\tau$ is invertible we can assume $\tau^k \beta = \beta$ for some particular $k$.
Taking the least common multiple of all such $k$ yields a uniform $k$ that works for all $\beta$ simultaneously, so $\tau^k \beta = \beta$ for all $\beta \in \Phi$.
Since $\RR\Phi = \EE, \tau^k$ acts as $\id$ on all of $\EE$, so $\tau^k - 1 = 0$ and so $m_\tau(t) \divides t^k - 1$ for some $k$.
Therefore $m_\tau(t) \divides \gcd( (t-1)^2, t^k-1 ) = t-1$, forcing $\tau = \id$ and $\sigma = s_ \alpha$ and $P = P_\alpha$.
:::

## Abstract root systems

:::{.definition title="Root systems"}
A subset $\Phi \subseteq \EE$ of a real Euclidean space is a **root system** iff

- R1: $\size \Phi < \infty$, $\RR\Phi = \EE$, and $0\not\in \Phi$,
- R2: $\alpha\in \Phi \implies -\alpha\in \Phi$ and no other scalar multiples of $\alpha$ are in $\Phi$,
- R3: If $\alpha\in \Phi$ then $s_\alpha( \Phi) = \Phi$,
- R4: If $\alpha, \beta\in \Phi$ then \( (\beta, \alpha\dual) = {2(\beta, \alpha) \over ( \alpha, \alpha) } \in \ZZ \).

Notably, $\ell\da \norm{\beta - s_\alpha\beta}$ is an integer multiple of $\alpha$:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/LieAlgebras/sections/figures}{2022-10-05_09-33.pdf_tex} };
\end{tikzpicture}

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:::{.definition title="Weyl groups"}
The **Weyl group** $W$ associated to a root system $\Phi$ is the subgroup $\gens{s_\alpha, \alpha\in \Phi} \leq \GL(\EE)$.
:::

:::{.remark}
Note that $\size W < \infty$: $W$ permutes $\Phi$ by (R3), so there is an injective group morphism $W \injects \mathrm{Perm}(\Phi)$, which is a finite group -- this is injective because if $w\actson \Phi$ as $\id$, since $\RR \Phi = \EE$, by linearity $w\actson \EE$ by $\id$ and $w=\id$.
Recalling that $s_ \alpha( \lambda) = \lambda- (\lambda, \alpha\dual) \alpha$, we have $(s_ \alpha(\lambda), s_ \alpha(\mu)) = ( \lambda, \mu)$ for all $\lambda, \mu \in \EE$.
So in fact $W \leq \Orth(\EE) \leq \GL(\EE)$, which have determinant $\pm 1$ -- in particular, $\det s_\alpha = -1$ since it can be written as a block matrix $\diag(1, 1, \cdots, 1, -1)$ by choosing a basis for $P_\alpha$ and extending it by $\alpha$.

> Note that one can classify finite subgroups of $\SO_n$.

:::

:::{.example title="?"}
Let $\Phi = \ts{ \eps_i - \eps_j \st 1\leq i,j \leq n+1, i\neq j}$ be a root system of type $A_n$ where $\ts{\eps_i}$ form the standard basis of $\RR^{n+1}$ with the standard inner product, so $(\eps_i, \eps_j) = \delta_{ij}$.
One can compute
\[
s_{\eps_i - \eps_j}(\eps_k) = \eps_k {2 (\eps_k, \eps_i - \eps_j) \over (\eps_i - \eps_j, \eps_i - \eps_j)}(\eps_i - \eps_j)
= \eps_k - (\eps_k, \eps_i - \eps_j)(\eps_i - \eps_j) 
=
\begin{cases}
\eps_j & k=i 
\\
\eps_i & k=j
\\
\eps_k & \text{otherwise}.
\end{cases}
=
\eps_{(ij).k}
\]
where $(ij) \in S_{n+1}$ is a transposition, acting as a function on the index $k$. 
Thus there is a well-defined group morphism
\[
W &\to S_{n+1} \\
s_{\eps_i - \eps_j} &\mapsto (ij)
.\]
This is injective since $w$ acting by the identity on every $\eps_k$ implies acting by the identity on all of $\EE$ by linearity, and surjective since transpositions generate $S_{n+1}$.
So $W\cong S_{n+1}$, and $A_n$ corresponds to $\liesl_{n+1}(\CC)$ using that
\[
[h, e_{ij}] = (h_i - h_j) e_{ij} = (\eps_i - \eps_j)(h) e_{ij}
.\]
In $G = \SL_n(\CC)$ one can define $N_G(T)/C_G(T)$ for $T$ a maximal torus.
:::

:::{.exercise title="?"}
What are the Weyl groups of other classical types?
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:::{.lemma title="?"}
Let $\Phi \subseteq \EE$ be a root system.
If $\sigma\in \GL(\EE)$ leaves $\Phi$ invariant, then for all $\alpha\in \Phi$, 
\[
\sigma s_{ \alpha} \sigma = s_{ \sigma( \alpha)}, \qquad (\beta, \alpha\dual) = (\sigma( \beta), \sigma(\alpha)\dual) \,\,\forall \alpha, \beta\in \Phi
.\]
Thus conjugating a reflection yields another reflection.
:::

:::{.proof title="?"}
Note that \( \sigma s_ \alpha \sigma\inv \) sends \( \sigma( \alpha) \) to its negative and fixes pointwise the hyperplane $\sigma(P_\alpha)$.
If \( \beta \in \Phi\) then \(s_{ \alpha}( \beta) \in \Phi \), so \( \sigma s_ \alpha ( \beta) \in \Phi \) and 
\[
(\sigma s_ \alpha \sigma\inv) ( \sigma( \beta)) = \sigma s_ \alpha(\beta) \in \sigma\Phi
,\]
so \( \sigma s_ \alpha \sigma\inv \) leaves invariant the set $\ts{ \sigma( \beta) \st \beta\in \Phi} = \Phi$.
By the previous lemma, it must equal $s_{ \sigma( \alpha)}$, and so 
\[
( \sigma( \beta), \sigma( \alpha)\dual ) = (\beta, \alpha\dual)
\]
by applying both sides to $\sigma(\beta)$.
:::

:::{.warnings}
This does not imply that \( (\sigma( \beta), \sigma( \alpha) ) = (\beta, \alpha) \)! 
With the duals/checks, this bracket involves a ratio, which is preserved, but the individual round brackets are not.
:::