---- Let $D(\cdot)$ be the assignment of a [[root system]] to its [[Dynkin diagram]]. > [!theorem] Theorem. ([[classification of irreducible root systems]]) > The association $\Phi \mapsto D(\Phi)$ induces a [[bijection]] between isomorphism classes of [[reducible root system|irreducible root systems]] and the following: ![[Pasted image 20250423121941.png]] > The four infinite families $A_{n},B_{n},C_{n},D_{n}$ are called the **classical root systems/Dynkin diagrams**, and the five remaining cases are caleld the **exceptional root systems**. $E_{8}$ is the most exceptional of all. > Can remember the classical Dynkin diagrams by remembering that $A,B,C$ are nominally similar but $A$ is [[simply laced root system|simply laced]] (so no orientations) and $C$ ends with an orientation 'lt; (which looks like a $C$). In contrast, $D$ **D**iverges at its end. > [!NOTE] Notes. > - If we want to avoid duplicates in the above list, we should assume $n \geq 2$ in the $B_{n}$ case, $n \geq 3$ in the $C_{n}$ case, and $n \geq 4$ in the $D_{n}$ case. > - Mnemonic: The <'$ in $C_{n}$ looks like a C'$. The >'$ in $B_{n}$ looks a bit like a '$B. > [!proof]- Proof. ([[classification of irreducible root systems]]) > Lots of combinatorics and Euclidean geometry, which for reasons of time won't be done in this course. See Humphrey's 11.4. ---- #### ----- #### 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 > ```