# Weyl Groups, 1.3 (Wednesday, September 01)

:::{.remark}
We'll spend a few days discussing Weyl groups, since they're important in the study of Schubert varieties.
For other references, see

- Björner and Brenti: *Combinatorics of Coxeter Groups*
:::

## Root Systems

:::{.remark}
Recall that given a generalized Cartan matrix $A$, there is an associated realization $(\lieh, \pi \subseteq \lieh\dual, \pi\dual)$.
:::

:::{.definition title="Reflections"}
For $1\leq i \leq \ell$, define a **reflection** $s_i\in \Aut(\lieh\dual)$ as
\[
s_i(\chi) \da \chi - \inner{\chi}{\alpha_i\dual}\alpha_i && \forall \chi \in \lieh\dual
.\]
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:::{.remark}
One can check that this fixes a hyperplane, and $s_i^2 = \id$.
:::

:::{.definition title="Crystallographic Root Systems"}
A subset $\Phi$ of Euclidean space $(V, \inner{\wait}{\wait})$ is a **crystallographic root system** in $V$ iff

1. $\Phi$ is finite, $\spanof_\RR \Phi = V$, and $0\not\in\Phi$.
2. If $\alpha \in \Phi$, then $\RR\alpha \intersect \Phi = \pm \alpha$.
3. If $\alpha \in \Phi$, then $s_\alpha$ leaves $\Phi$ invariant
4. If \( \alpha, \beta\in \Phi \), then ${(\beta, \alpha) \over 2(\alpha, \alpha)} \in \ZZ$.
:::

:::{.remark}
Note that for a Kac Moody Lie algebra, $\Phi$ is often infinite, so condition 1 can fail.
Condition 2 can fail if $\alpha$ is imaginary, in which case $n\alpha \in \Phi$ for some $n\in \ZZ$.
:::

:::{.definition title="Weyl Groups"}
Let $W \subseteq \Aut(\lieh\dual)$ be the subgroup generated by $\ts{s_i \st 1\leq i\leq \ell}$, then $W$ is said to be the **Weyl group** of $\lieg$.
:::

:::{.definition title="Lengths"}
Let $\mcw$ be the group generated by a fixed set $S$ of elements of order 2 in $W$.
Then for $w\in \mcw$, the **length** $\ell(w)$ is the smallest number $\ell$ such that $w = \prod_{i=1}^\ell s_i$.
:::

:::{.remark}
Note that $\ell(1) = 0$, and for $Y \subseteq S$, we set $\mcw_Y$ to be the subgroup generated by $\ts{s\st s\in Y}$.
We'll prove that any Weyl group is a Coxeter group, but for now $W$ is a Weyl group and $\mcw$ is a Coxeter group.
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:::{.theorem title="1.3.11"}
Let $(\mcw, S)$ be as above, then TFAE:

1. The Coxeter condition: 
  $\mcw$ is a quotient of the free group $\hat \mcw$ generated by $S$, modulo the following relations:

  - $s^2= 1$ for all $s\in S$.
  - $(st)^{m_{s, t}} = 1$ for all $s\neq t$ in $S$ and for some integers $m_{t, s} = m_{s, t} \geq 2$ (or possibly $\infty$).

2. The root system condition:
  There exists a representation $V$ of $\mcw$ over $\RR$ together with a subset $\Delta \subseteq V\smts{0}$ such that

  - Symmetric: $\Delta = -\Delta$
  - $\mcw\dash$invariance/stability: there exists a subset $\pi \da \ts{\alpha_s}_{s\in S} \subseteq \Delta$ such that for any \( \alpha \in \Delta \) exactly one of $\alpha$ or $-\alpha$ belongs to the set of positive linear combinations of "simple roots" $\sum_{s\in S} \RR_{>0} \alpha_s$.
    If $\alpha$ is in this subset, we'll say $\alpha$ is **positive**, and if $-\alpha$ is in it, we'll say $\alpha$ is **negative**.
  - For every $s\in S$, if $\alpha \neq \alpha_s$ and $\alpha > 0$ is positive, then $s\alpha_s <0$ is negative and $s\alpha > 0$.
  [^makespositive]

  - For $s, t\in S$ and $w\in \mcw$, then $w \alpha_s = \alpha_t$ implies that $wsw\inv = t$, so the group action is captured in a conjugation action.

3. The *strong exchange* condition:
  For $s\in S$ and $v, w\in \mcw$ with $\ell(vsv\inv w) \leq \ell(w)$, for any expression $w = \prod_{i=1}^n s_i$ with $s_i \in S$, we have $vsv\inv w = \prod_{i\neq j}^n s_i$ for some $j$.

4. The *exchange condition*:
  For $s\in S, w\in \mcw$ with $\ell(sw) \leq  \ell(w)$, then for any reduced expression $w = \prod_{i=1}^n s_i$, we have $sw = \prod_{i\neq j}^n s_i$ for some $j$.

  [^makespositive]: 
  So the simple reflection changes the sign of only the corresponding simple root, and preserves the sign of all other simple roots.
  
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:::{.remark}
These conditions show up in a lot of proofs!
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:::{.definition title="Crystallographic Coxeter groups"}
If $S$ is finite (which it will be for us), we can take $V$ to be finite dimensional.
Writing $S \da \ts{s_1, \cdots, s_\ell}$ and set $m_{ij} \da \order(s_i s_j)$.
If every $m_{ij}$ is one of $\ts{2,3,4,6,\infty}$, call the Coxeter group **crystallographic**.
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:::{.remark}
An open problem is that all Coxeter groups *should* come from geometry, e.g. from projective varieties (?), but it's not clear what these varieties should be.
The crystallographic ones will precisely come from Kac-Moody Lie algebras.
This is closely related to problems concerning KL polynomials: take an Ind variety, stratify it, and take intersection cohomology.
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:::{.remark}
Every *finite* irreducible Coxeter group (with exceptions $H_3, H_4, I_2(m)$) occur as Weyl groups of crystallographic root systems.
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:::{.proof title="of theorem, $1\implies 2$"}
Let $V$ be the $\RR\dash$module with basis $\ts{\alpha_s \st s\in S}$.
For any $s\in S$, define an inner product by extending the following $\RR\dash$bilinearly:
\[
(\alpha_s, \alpha_s) &= 1 \\
(\alpha_{s_1}, \alpha_{s_2}) &= \cos({\pi \over m_{s_1s_2}}) && s_1\neq s_2 
.\]

For $s, v\in V$, define
\[
s(v) \da v - 2(v, \alpha_s) \alpha_s
.\]
A quick computation shows
\[
s( \alpha_s) = \alpha_s - 2 (\alpha_s, \alpha_s) \alpha_s = - \alpha_s
.\]
One can check that the formula is $\RR\dash$linear, and using this we have
\[
s^2(v) 
&= s(v - 2(v, \alpha_s) \alpha_s) \\
&= s(v) - 2(v, \alpha_s) s(\alpha_s) \\
&= (v -2(v, \alpha_s) \alpha_s) - 2(v, \alpha_s)s( \alpha_s) \\
&= (v -2(v, \alpha_s) \alpha_s) - 2(v, \alpha_s)(- \alpha_s) \\
&= v
,\]
so $s^2 = \id$.
By assumption, we have $(s_1 s_2)^{m_{s_1 s_2}}(v) = v$.
Using that this formula factors through the relations, we can extend this to an action $\mcw \actson V$.
Then
\[
(s(v), s(v')) 
&= (v - 2(v, \alpha_s) \alpha_2, v' - 2(v', \alpha_s) \alpha_s) \\
&= (v, v') - 2(v', \alpha_s)(v, \alpha_s) - 2(v, \alpha_s)(\alpha_s, v') + 4(v, \alpha_s)(v', \alpha_s)(\alpha_s, \alpha_s) \\
&= (v, v') - 4(v', \alpha_s)(v, \alpha_s) + 4(v, \alpha_s)(v', \alpha_s)\\
&= (v, v')
,\]
where we've used that $(\wait, \wait)$ is symmetric.
Thus $(wv, wv') = (v, v')$.

Let \( \Delta\da \Union_{s\in S} \mathcal{W} ( \alpha_s) \), we'll work with this more next time.
:::