---- > [!definition] Definition. ([[Dynkin diagram]]) > Let $(\Phi, E)$ be a [[root system]], $\Delta$ a [[root basis]]. The corresponding **Dynkin diagram** is the decorated [[network|graph]] $D(\Phi)$ defined by the properties[[reflection|:]] > - Vertices are [[root basis|simple roots]] $\alpha_{1},\dots,\alpha_{\ell}$ > - Two (distinct)vertices $\alpha, \beta$ are connected by an edge iff $(\alpha, \beta) \neq 0$. In that case, we draw exactly $\underbrace{ \langle \alpha, \check \beta \rangle \langle \beta, \check \alpha \rangle }_{ \in \{ 1,2,3 \} }$ edges between them > - Suppose $\alpha, \beta \in \Delta$ are connected by an edge. If $(\alpha, \alpha) = (\beta, \beta)$, we declare the edge to be undirected. If the two roots have different length, $(\alpha, \alpha)< (\beta, \beta)$[^1], we capture this by drawing an 'lt; on the edge from $\alpha$ to $\beta$. > > The [[network|unoriented]] graph underlying the Dynkni diagram is called the **Coxeter graph** of $\Phi$. > $\Phi$ is [[reducible root system|irreducible]] [[root system is irreducible iff Dynkin diagram is connected|if and only if]] its Dynkin diagram $D(\Phi)$ is [[connected]]. > [!basicexample] (Rank 1 and Rank 2 Dynkin Diagrams) > Here are the Dynkin diagrams for the rank 1 and 2 root systems: > ![[Pasted image 20250423115401.png]] > ![[Pasted image 20250423115433.png]] where in the cases $B_{2},G_{2}$, the leftmost vertex corresponds to the long root in the [[root basis]]. > Note that the [[cartan matrix|Cartan matrix]] readily specifies how many edges should be present between a pair of nodes. The [[root basis]] readily specifies everything else. ^basic-example > [!basicexample] (Dynkin Diagram for $A_{n}$) > Consider the $A_{n}$ [[root system]] $\Phi=\{ e_{i}-e_{j}: 1 \leq i,j \leq n+1 \}$. We know it has [[root basis]] $\Delta=\{ e_{1}-e_{2}, \dots, e_{n}-e_{n+1} \}$. The Dynkin diagram is > ![[Pasted image 20250423121644.png]] More examples can be found in the [[classification of irreducible root systems]]. ^basic-example ---- #### [^1]: [[angles and lengths in a root system|This happens exactly when]] $\langle \beta, \alpha^{\vee} \rangle \in \{ -2, -3 \}$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```