# Wednesday October 9

**Last time:**

We have the Weyl group $W \coloneqq \theset{S_\alpha \suchthat \alpha \in \Phi} = \theset{S_\alpha \suchthat S_\alpha \in \Pi}$.
If $W \ni w = \prod{i=1}^\ell W_{\alpha_i}$ is a product of simple reflections, then $W$ is said to be *reduced* if $\ell$ is the smallest among all such products.
Call $\ell(w)$ the length of $W$ and let $n(W) = \# N_W$.
By Corollary 10.2b, $N_W = \theset{\alpha \in \Phi^+ \suchthat W(\alpha) \in \Phi^-}$, and if $W = \prod S_{\alpha_i}$ is reduced, then $w(\alpha_j) \in \Phi^-$.

**Lemma:** 
$\ell(w) = n(w)$.

*Proof:*
Done in class, but see Humphrey's.

## Classification

### Cartan Matrix

Fix a base $\Pi \subset \Phi$ of rank $\ell$.

**Definition:**
Fix an order $(\alpha_1, \alpha_2, \cdots \alpha_\ell$ of $\Pi$. 
Then the *Cartan matrix* is given by $A_{ij} = \inner{\alpha_i}{\alpha_j\dual} \in \mathrm{Mat}(\ell\times\ell, \ZZ)$.

*Examples:*

![Image](figures/2019-10-09-09:24.png)\

![Image](figures/2019-10-09-09:26.png)\

**Facts:**

a. $A$ depends on the chosen ordering of $\Pi$.
b. $A$ is independent of the choice of $\Pi$.
c. $A$ is invertible.
d. $A$ uniquely determines the root system (up to isomorphism).
  I.e., if $A(\Phi) = A(\Phi')$ then there is an isomorphism $E \mapsvia{\phi} E$ on the underlying Euclidean space such that $\\phi(\Phi) = \Phi'$ and $\inner{\alpha}{\beta\dual} = \inner{\phi(\alpha)}{\phi(\beta)\dual}$ for all $\alpha, \beta \in \Phi$.


### Dynkin Diagrams

Recall from Lemma 9.4 that $a_{ij} a_{ji} \in \theset{0,1,2,3}$.

**Definition:**
Given a Cartan matrix $A$, its Coxeter diagram is an undirected multigraph $\Gamma = (I, E)$  where $I$ is a vertex set and the edge set is given by edges between vertices corresponding to $i, j$ (where $i\neq j$) with weight $a_{ij} a_{ji}$.

*Examples:*

Note that these diagrams don't encode which roots are longer, so we can decorate these diagrams with arrows to indicate this and obtain a partially-directed multigraph.

**Definition:**
A *Dynkin diagram* is the partially-directed multigraph obtained from the Coxeter diagram by adding arrows on the double or triple edges between $i, j$ precisely when $\abs{a_i} > \abs{a_j}$.
(Note that this also occurs when $\abs a_{ij} < \abs a_{ji}$)


**Definition:**
A non-empty root system is *irreducible* if $\Phi \neq \Phi_1 \oplus \Phi_2$ for some nonempty root system $\Phi_2$ where $\alpha\in \Phi_1, \beta\in \Phi_2 \implies \inner{\alpha}{\beta} = 0$.

For example: $\Phi(A_1 \cross A_1)$ can be written as $\Phi(A_1) \oplus \phi(A_1)$ since the off-diagonal entries were zero, so it is reducible.

**Facts:**

a. $\Phi$ is irreducible iff the Dynkin diagram is connected
b. $\Phi$ can be uniquely written as the union of irreducible root systems (where the multiplicity of each system appearing is well-defined)


Thus to classify root systems, it suffices to classify connected Dynkin diagrams.

*Examples of Dynkin diagrams:*

![Image](figures/2019-10-09-21:03.png)

![Image](figures/2019-10-09-21:03.png)