# 루트 시스템 (root system)과 딘킨 다이어그램 (Dynkin diagram)

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## 정의

• E를 내적이 주어진 유클리드 벡터공간이라 하자.
• 다음 조건을 만족시키는 E의 유한인 부분집합 $$\Phi$$를 루트 시스템이라 한다.
• $$\Phi$$는 E를 스팬(span)하며 $$0 \not \in \Phi$$
• (reduced) $$\alpha \in \Phi$$, $$\lambda \alpha \in \Phi \iff \lambda=\pm 1$$
• $$\alpha,\beta \in \Phi$$이면 $$\sigma_\alpha(\beta) =\beta-2\frac{(\beta,\alpha)}{(\alpha,\alpha)}\alpha \in \Phi$$
• $$\langle \beta, \alpha \rangle = 2 \frac{(\beta,\alpha)}{(\alpha,\alpha)} \in \mathbb{Z}$$
• 마지막 조건을 crystallographic 또는 integrality 조건이라 한다
• a subgroup of $$GL(V)$$ is crystallographic if it stabilizes a lattice L in V
• e.g. the Weyl group of a Lie algebra stabilizes the root lattice or the weight lattice

## 딘킨 다이어그램 (Dynkin diagram)

• first draw the simple roots as nodes
• draw $$4(e_i, e_j)^2$$lines for two roots $$e_i, e_j$$
• $$\frac{\pi}{2}$$ , $$\frac{\pi}{3}$$, $$\frac{\pi}{4}$$, $$\frac{\pi}{6}$$ 0,1,2,3 lines

## 2차원 루트 시스템의 분류

• $$A_1\times A_1$$, $$A_2$$, $$B_2$$, $$G_2$$

A1 x A1

A2

B2

G2

## ADE 의 분류

(0) G cannot contain affine A_n, D_n, E_n

(1) G is a tree (contains no cycles = affine A_n)

(2) G has \leq 1 branch point (does not contain affine D_5, D_6,D_7, )

(3) branch point has order \leq 3 (affine D_4) What are length of legs of G?

Leg of length 0 -> G=A_n

so assume legs have length \geq 1

(4) Not all legs have length \geq 2 : cannot contain affine E_6

so one leg has length 1

2 legs of length 1 : G is D_n

so can assume 2 other legs have length \geq 2

(5) cannot have 2 legs length \geq 3 because of affine E_7

So G has 1 leg length 1, 1 of length 2, one of length \geq 2

length is \leq 4, as G does not contain affine E_8

So G is E6,E7, E8

일반적인 경우

• how to classify all connected admissible diagrams
• subdiagram is also admissible
• there are at most (n-1) pairs of nodes
• no node has more than 3 lines
• study double lines and triple nodes

## reflection groups

• B_n, C_n, BC_n -> same reflection group (Z/nZ).S_n