# Friday, October 21
## 11.1: The Cartan Matrix
:::{.definition title="Cartan matrix"}
Fix $\Delta \subseteq \Phi$ a rank $\ell$ root system with Weyl group $W$.
Let $\Delta = \tsl \alpha 1 \ell$ and then the matrix $A$ where $A_{ij} = \inp {\alpha_i}{\alpha_j} = 2{\inp{\alpha_i}{\alpha_j} \over \inp{\alpha_j}{\alpha_j}}$ is the **Cartan matrix of $A$.**
Note that changing the ordering of $\Delta$ permutes the rows and columns of $A$, but beyond this, $A$ does not depend on the choice of $\Delta$ since they are permuted by $W$ and $W$ preserves the inner products and thus the ratios defining the *Cartan numbers* $A_{ij}$.
More $A\in \GL_\ell(\ZZ)$ since the inner product is nondegenerate and $\Delta$ is a basis for $\EE$.
:::
:::{.example title="?"}
Note that the diagonals are always 2.
Some classical types:
- $A_1 \times A_1: \matt 2002$
- $A_2: \matt 2 {-1} {-1} 2$
- $B_2: \matt 2 {-2}{-1} 2$
- $G_2: \matt 2 {-1}{-3} 2$.
:::
:::{.remark}
The Cartan matrix $A$ determines the root system $\Phi$ up to isomorphism: if $\Phi' \subseteq \EE'$ is another root system with base $\Delta' = \tsl{ \alpha'} 1 \ell$ with $A'_{ij} = A_{ij}$ for all $i, j$ then the bijection $\alpha_i \mapsto \alpha_i'$ extends to a bijection $\phi: \EE \iso \EE'$ sending $\Phi$ to $\Phi'$ which is an isometry, i.e. $\inp{\varphi(\alpha)}{\varphi( \beta)} = \inp \alpha \beta$ for all \( \alpha, \beta \in \Phi \).
Since $\Delta, \Delta'$ are bases of $\EE$, this gives a vector space isomorphism $\phi(\alpha_i) \da \alpha_i'$.
If $\alpha, \beta\in \Delta$ are simple, then
\[
s_{\varphi( \alpha)}( \varphi( \beta))
&= \varphi( \beta)- \inp{\beta'}{\alpha'}\phi( \alpha) \\
&= \varphi( \beta)-\inp{ \beta}{ \alpha} \phi( \alpha) \\
&= \phi(\beta- \inp \beta \alpha \alpha) \\
&= \phi( s_ \alpha( \beta))
,\]
so this diagram commutes since these maps agree on the simple roots, which form a basis:
\begin{tikzcd}
\EE && \EE \\
\\
\EE && \EE
\arrow["\phi", from=1-1, to=1-3]
\arrow["{s_\alpha}", from=1-3, to=3-3]
\arrow["\phi"', from=3-1, to=3-3]
\arrow["{s_\alpha}"', from=1-1, to=3-1]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXEVFIl0sWzIsMCwiXFxFRSJdLFswLDIsIlxcRUUiXSxbMiwyLCJcXEVFIl0sWzAsMSwiXFxwaGkiXSxbMSwzLCJzX1xcYWxwaGEiXSxbMiwzLCJcXHBoaSIsMl0sWzAsMiwic19cXGFscGhhIiwyXV0=)
Since $W, W'$ are generated by reflections and $s_{ \varphi( \alpha)} = \varphi\circ s_ \alpha \circ \varphi\inv$ for $\alpha\in \Delta$, there is an isomorphism
\[
W &\iso W \\
s_ \alpha &\mapsto s_{ \varphi( \alpha)} = \varphi s_ \alpha \varphi \quad \forall \alpha \in \Delta
.\]
If $\beta \in \Phi$, then $\beta = w( \alpha)$ for some $\alpha\in \Delta$ and $w\in W$ by theorem 10.3C.
Thus $\phi( \beta) = ( \varphi \circ w \circ \varphi\inv)( \varphi( \alpha))\in \Phi'$ since \( \varphi\circ w \circ \varphi\inv \in W' \).
Thus $\phi( \Phi) = \Phi'$.
Using lemma 9.2, $s_{\varphi(\beta)} = \varphi s_ \beta \varphi\inv$, so $\phi$ preserves all of the Cartan integers $\inp \beta \gamma$ for all \( \gamma, \beta\in \Phi \).
:::
:::{.remark}
Read the last paragraph of $\S 11.1$ which gives an algorithm for constructing $\Phi^+$ from $\Delta$ and $A$.
:::
## 11.2: Coxeter graphs and Dynkin diagrams
:::{.definition title="Coxeter graph"}
If $\alpha\neq \beta\in \Phi^+$ then $\inp \beta \alpha\inp \alpha \beta = 0,1,2,3$ from the table several sections ago.
Fix $\Delta = \tsl \alpha 1 \ell$, then the **Coxeter graph** $\Gamma$ of $\Phi$ is the graph with $\ell$ vertices $1,\cdots, \ell$ with vertices $i, j$ connected by $\inp {\alpha_i}{ \alpha_j} \inp {\alpha_j}{\alpha_i}$ edges.
:::
:::{.example title="?"}
Recall that the table was
| $\inp \alpha \beta$ | $\inp \beta \alpha$ |
|---|---|
| 0 | 0 |
| -1 | -1 |
| -1 | -2 |
| -1 | -3 |
Here $\alpha$ is the shorter root., although without loss of generality in the first two rows we can rescale so that $\norm \alpha= \norm \beta$.
The graphs for some classical types:
:::
:::{.remark}
If $\Phi$ has roots all of the same length, the Coxeter graph determines the Cartan integers since $A_{ij} = 0, 1$ for $i\neq j$.
If $i \to j$ is a subgraph of $\Gamma$ then $\inp{ \alpha_i}{ \alpha_j} = \inp{ \alpha_j}{\alpha_i} = -1$, so \( \alpha_i, \alpha_j \) have the same length.
However, if there are roots of multiple lengths, taking the product to determine the number of edges loses information about which root is longer.
:::
:::{.definition title="Dynkin diagram"}
The **Dynkin diagram** of $\Phi$ is the Coxeter graph $\Gamma$ where for each multiple edge, there is an arrow pointing from the longer root to the shorter root.
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:::{.example title="?"}
In rank 2:
We also have the following diagram for $F_4$:
:::
:::{.remark}
Note that $\Phi$ is irreducible iff $\Delta$ can not be partitioned into two proper nonempty orthogonal subsets iff the Coxeter graph is connected.
In general, if $\Gamma$ has $t$ connected components, let $\Delta = \Disjoint_{1\leq i\leq t} \Delta_i$ be the corresponding orthogonal partition of simple roots.
Let $\EE_i = \spanof_\RR\Delta_i$, then $\EE = \bigoplus_{1\leq i\leq t}\EE_i$ is an orthogonal direct sum decomposition into $W\dash$invariant subspaces, which follows from the reflection formula.
Writing $\Phi_i = (\ZZ \Delta_I) \intersect \Phi$, one has $\Phi = \Disjoint_{1\leq i\leq t} \Phi_i$ since each root is $W\dash$conjugate to a simple root and $\ZZ\Delta_i$ is $W\dash$invariant and each $\Phi_i \subseteq \EE_i$ is itself a root system.
Thus it's enough to classify irreducible root systems.
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