E8 루트 시스템

개요

• 다양한 수학적 대상이 E8 이라는 이름으로 불린다
  • 루트 시스템으로서의 E8
  • 정수 계수 이차형식으로서의 E8
  • 단순리대수로서의 E8
  • 단순리군으로서의 E8

딘킨 다이어그램

카르탄 행렬 \[ \left( \begin{array}{cccccccc} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 \\ \end{array} \right) \]

루트 시스템

• 240개의 8차원 벡터로 구성
• 테이블

\[ \begin{array}{c|c|c|c} \text{} & \text{height} & \text{label} & \text{coordinate} \\ \hline 1 & 1 & \{1,0,0,0,0,0,0,0\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 2 & 1 & \{0,1,0,0,0,0,0,0\} & \{-1,1,0,0,0,0,0,0\} \\ 3 & 1 & \{0,0,1,0,0,0,0,0\} & \{0,-1,1,0,0,0,0,0\} \\ 4 & 1 & \{0,0,0,1,0,0,0,0\} & \{0,0,-1,1,0,0,0,0\} \\ 5 & 1 & \{0,0,0,0,1,0,0,0\} & \{0,0,0,-1,1,0,0,0\} \\ 6 & 1 & \{0,0,0,0,0,1,0,0\} & \{0,0,0,0,-1,1,0,0\} \\ 7 & 1 & \{0,0,0,0,0,0,1,0\} & \{0,0,0,0,0,-1,1,0\} \\ 8 & 1 & \{0,0,0,0,0,0,0,1\} & \{1,1,0,0,0,0,0,0\} \\ 9 & 2 & \{0,0,1,0,0,0,0,1\} & \{1,0,1,0,0,0,0,0\} \\ 10 & 2 & \{0,0,0,0,0,1,1,0\} & \{0,0,0,0,-1,0,1,0\} \\ 11 & 2 & \{0,0,0,0,1,1,0,0\} & \{0,0,0,-1,0,1,0,0\} \\ 12 & 2 & \{0,0,0,1,1,0,0,0\} & \{0,0,-1,0,1,0,0,0\} \\ 13 & 2 & \{0,0,1,1,0,0,0,0\} & \{0,-1,0,1,0,0,0,0\} \\ 14 & 2 & \{1,1,0,0,0,0,0,0\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 15 & 2 & \{0,1,1,0,0,0,0,0\} & \{-1,0,1,0,0,0,0,0\} \\ 16 & 3 & \{0,0,1,1,0,0,0,1\} & \{1,0,0,1,0,0,0,0\} \\ 17 & 3 & \{0,1,1,0,0,0,0,1\} & \{0,1,1,0,0,0,0,0\} \\ 18 & 3 & \{0,0,0,0,1,1,1,0\} & \{0,0,0,-1,0,0,1,0\} \\ 19 & 3 & \{0,0,0,1,1,1,0,0\} & \{0,0,-1,0,0,1,0,0\} \\ 20 & 3 & \{0,0,1,1,1,0,0,0\} & \{0,-1,0,0,1,0,0,0\} \\ 21 & 3 & \{1,1,1,0,0,0,0,0\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 22 & 3 & \{0,1,1,1,0,0,0,0\} & \{-1,0,0,1,0,0,0,0\} \\ 23 & 4 & \{0,0,1,1,1,0,0,1\} & \{1,0,0,0,1,0,0,0\} \\ 24 & 4 & \{1,1,1,0,0,0,0,1\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 25 & 4 & \{0,1,1,1,0,0,0,1\} & \{0,1,0,1,0,0,0,0\} \\ 26 & 4 & \{0,0,0,1,1,1,1,0\} & \{0,0,-1,0,0,0,1,0\} \\ 27 & 4 & \{0,0,1,1,1,1,0,0\} & \{0,-1,0,0,0,1,0,0\} \\ 28 & 4 & \{1,1,1,1,0,0,0,0\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 29 & 4 & \{0,1,1,1,1,0,0,0\} & \{-1,0,0,0,1,0,0,0\} \\ 30 & 5 & \{0,0,1,1,1,1,0,1\} & \{1,0,0,0,0,1,0,0\} \\ 31 & 5 & \{1,1,1,1,0,0,0,1\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 32 & 5 & \{0,1,1,1,1,0,0,1\} & \{0,1,0,0,1,0,0,0\} \\ 33 & 5 & \{0,1,2,1,0,0,0,1\} & \{0,0,1,1,0,0,0,0\} \\ 34 & 5 & \{0,0,1,1,1,1,1,0\} & \{0,-1,0,0,0,0,1,0\} \\ 35 & 5 & \{1,1,1,1,1,0,0,0\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 36 & 5 & \{0,1,1,1,1,1,0,0\} & \{-1,0,0,0,0,1,0,0\} \\ 37 & 6 & \{0,0,1,1,1,1,1,1\} & \{1,0,0,0,0,0,1,0\} \\ 38 & 6 & \{1,1,1,1,1,0,0,1\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 39 & 6 & \{1,1,2,1,0,0,0,1\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 40 & 6 & \{0,1,1,1,1,1,0,1\} & \{0,1,0,0,0,1,0,0\} \\ 41 & 6 & \{0,1,2,1,1,0,0,1\} & \{0,0,1,0,1,0,0,0\} \\ 42 & 6 & \{1,1,1,1,1,1,0,0\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 43 & 6 & \{0,1,1,1,1,1,1,0\} & \{-1,0,0,0,0,0,1,0\} \\ 44 & 7 & \{1,1,1,1,1,1,0,1\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 45 & 7 & \{1,1,2,1,1,0,0,1\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 46 & 7 & \{0,1,1,1,1,1,1,1\} & \{0,1,0,0,0,0,1,0\} \\ 47 & 7 & \{0,1,2,1,1,1,0,1\} & \{0,0,1,0,0,1,0,0\} \\ 48 & 7 & \{0,1,2,2,1,0,0,1\} & \{0,0,0,1,1,0,0,0\} \\ 49 & 7 & \{1,2,2,1,0,0,0,1\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 50 & 7 & \{1,1,1,1,1,1,1,0\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 51 & 8 & \{1,1,1,1,1,1,1,1\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 52 & 8 & \{1,1,2,1,1,1,0,1\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 53 & 8 & \{1,1,2,2,1,0,0,1\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 54 & 8 & \{0,1,2,1,1,1,1,1\} & \{0,0,1,0,0,0,1,0\} \\ 55 & 8 & \{0,1,2,2,1,1,0,1\} & \{0,0,0,1,0,1,0,0\} \\ 56 & 8 & \{1,2,2,1,1,0,0,1\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 57 & 9 & \{1,1,2,1,1,1,1,1\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 58 & 9 & \{1,1,2,2,1,1,0,1\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 59 & 9 & \{0,1,2,2,1,1,1,1\} & \{0,0,0,1,0,0,1,0\} \\ 60 & 9 & \{0,1,2,2,2,1,0,1\} & \{0,0,0,0,1,1,0,0\} \\ 61 & 9 & \{1,2,2,1,1,1,0,1\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 62 & 9 & \{1,2,2,2,1,0,0,1\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 63 & 10 & \{1,1,2,2,1,1,1,1\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 64 & 10 & \{1,1,2,2,2,1,0,1\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 65 & 10 & \{0,1,2,2,2,1,1,1\} & \{0,0,0,0,1,0,1,0\} \\ 66 & 10 & \{1,2,2,1,1,1,1,1\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 67 & 10 & \{1,2,2,2,1,1,0,1\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 68 & 10 & \{1,2,3,2,1,0,0,1\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 69 & 11 & \{1,2,3,2,1,0,0,2\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 70 & 11 & \{1,1,2,2,2,1,1,1\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 71 & 11 & \{0,1,2,2,2,2,1,1\} & \{0,0,0,0,0,1,1,0\} \\ 72 & 11 & \{1,2,2,2,1,1,1,1\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 73 & 11 & \{1,2,2,2,2,1,0,1\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 74 & 11 & \{1,2,3,2,1,1,0,1\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 75 & 12 & \{1,2,3,2,1,1,0,2\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 76 & 12 & \{1,1,2,2,2,2,1,1\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 77 & 12 & \{1,2,2,2,2,1,1,1\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 78 & 12 & \{1,2,3,2,1,1,1,1\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 79 & 12 & \{1,2,3,2,2,1,0,1\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 80 & 13 & \{1,2,3,2,1,1,1,2\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 81 & 13 & \{1,2,3,2,2,1,0,2\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 82 & 13 & \{1,2,2,2,2,2,1,1\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 83 & 13 & \{1,2,3,2,2,1,1,1\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 84 & 13 & \{1,2,3,3,2,1,0,1\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 85 & 14 & \{1,2,3,2,2,1,1,2\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 86 & 14 & \{1,2,3,3,2,1,0,2\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 87 & 14 & \{1,2,3,2,2,2,1,1\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 88 & 14 & \{1,2,3,3,2,1,1,1\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 89 & 15 & \{1,2,3,2,2,2,1,2\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 90 & 15 & \{1,2,3,3,2,1,1,2\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 91 & 15 & \{1,2,4,3,2,1,0,2\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 92 & 15 & \{1,2,3,3,2,2,1,1\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 93 & 16 & \{1,2,3,3,2,2,1,2\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 94 & 16 & \{1,2,4,3,2,1,1,2\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 95 & 16 & \{1,3,4,3,2,1,0,2\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} \\ 96 & 16 & \{1,2,3,3,3,2,1,1\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 97 & 17 & \{1,2,3,3,3,2,1,2\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 98 & 17 & \{1,2,4,3,2,2,1,2\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 99 & 17 & \{2,3,4,3,2,1,0,2\} & \{0,0,0,0,0,0,-1,1\} \\ 100 & 17 & \{1,3,4,3,2,1,1,2\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 101 & 18 & \{1,2,4,3,3,2,1,2\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 102 & 18 & \{2,3,4,3,2,1,1,2\} & \{0,0,0,0,0,-1,0,1\} \\ 103 & 18 & \{1,3,4,3,2,2,1,2\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 104 & 19 & \{1,2,4,4,3,2,1,2\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 105 & 19 & \{2,3,4,3,2,2,1,2\} & \{0,0,0,0,-1,0,0,1\} \\ 106 & 19 & \{1,3,4,3,3,2,1,2\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 107 & 20 & \{2,3,4,3,3,2,1,2\} & \{0,0,0,-1,0,0,0,1\} \\ 108 & 20 & \{1,3,4,4,3,2,1,2\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 109 & 21 & \{2,3,4,4,3,2,1,2\} & \{0,0,-1,0,0,0,0,1\} \\ 110 & 21 & \{1,3,5,4,3,2,1,2\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 111 & 22 & \{1,3,5,4,3,2,1,3\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} \\ 112 & 22 & \{2,3,5,4,3,2,1,2\} & \{0,-1,0,0,0,0,0,1\} \\ 113 & 23 & \{2,3,5,4,3,2,1,3\} & \{1,0,0,0,0,0,0,1\} \\ 114 & 23 & \{2,4,5,4,3,2,1,2\} & \{-1,0,0,0,0,0,0,1\} \\ 115 & 24 & \{2,4,5,4,3,2,1,3\} & \{0,1,0,0,0,0,0,1\} \\ 116 & 25 & \{2,4,6,4,3,2,1,3\} & \{0,0,1,0,0,0,0,1\} \\ 117 & 26 & \{2,4,6,5,3,2,1,3\} & \{0,0,0,1,0,0,0,1\} \\ 118 & 27 & \{2,4,6,5,4,2,1,3\} & \{0,0,0,0,1,0,0,1\} \\ 119 & 28 & \{2,4,6,5,4,3,1,3\} & \{0,0,0,0,0,1,0,1\} \\ 120 & 29 & \{2,4,6,5,4,3,2,3\} & \{0,0,0,0,0,0,1,1\} \end{array} \]

메모

• 정이십면체와 E8 (via Mckay correspondence)

관련된 항목들

• Kissing number and sphere packings
• E8 격자

관련된 학부 과목과 미리 알고 있으면 좋은 것들

• 선형대수학

관련된 대학원 과목

• 이차형식
• 리군과 리대수
• 리대수 g2의 유한차원 표현론

계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxZFZVdFFrMkJDWHc/edit

사전 형태의 자료

• http://en.wikipedia.org/wiki/E8

메모

• The E8 root system
• Borcherds' lectures on Lie groups from Anton's webpage[파일:1924972-]
  • Borcherds_on_LieGroups.pdf
• Bertram Kostant (Baez's webpage)[1]
  • On Some Mathematics in Garrett Lisi's 'E8 Theory of Everything'
• John Baez
  • My Favorite Numbers : 5, 8, and 24

리뷰, 에세이, 강의노트

• 피타고라스의 창
  • E8이란 무엇인가 (1) : 들어가며
  • E8이란 무엇인가 (2) : 8차원에서 내려온 그림자
  • E8이란 무엇인가 (3) : 8차원의 눈꽃송이
  • E8이란 무엇인가 (번외편) - E8과 모뎀

관련논문

• Dechant, Pierre-Philippe. “The Birth of \(E_8\) out of the Spinors of the Icosahedron.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 472, no. 2185 (January 2016): 20150504. doi:10.1098/rspa.2015.0504.

메타데이터

위키데이터

• ID : Q298759