■リー群と表現(その65)
【1】E8無限鏡映群の基本単体の頂点座標
(0,0,0,0,0,0,0,0)
(0,0,0,0,0,0,1/2,1/2)
(0,0,0,0,0,1/3,1/3,2/3)
(0,0,0,0,1/4,1/4,1/4,3/4)
(0,0,0,1/5,1/5,1/5,1/5,4/5)
(0,0,1/6,1/6,1/6,1/6,1/6,5/6)
(1/6,1/6,1/6,1/6,1/6,1/6,1/6,5/6)
(−1/8,1/8,1/8,1/8,1/8,1/8,1/8,7/8)
(0,0,0,0,0,0,0,1)
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【2】E8のボロノイ細胞
[1]x7+x8=1
[2]x1+x8=x2+x3+・・・+x7
[3]x1=x2
[4]x2=x3
[5]x3=x4
[6]x4=x5
[7]x5=x6
[8]x6=x7
[9]x1=−x2
[2]を外すと
x1=x2=x3=x4=x5=x6=x7=0
x8=1
[3]を外すと
x1=1/8
x2=x3=x4=x5=x6=x7=1/8
x8=7/8
[4]を外すと
x1=x2=0
x3=x4=x5=x6=x7=1/6
x8=5/6
[5]を外すと
x1=x2=x3=0
x4=x5=x6=x7=1/5
x8=4/5
[6]を外すと
x1=x2=x3=x4=0
x5=x6=x7=1/4
x8=3/4
[7]を外すと
x1=x2=x3=x4=x5=0
x6=x7=1/3
x8=2/3
[8]を外すと
x1=x2=x3=x4=x5=x6=0
x7=1/2,x8=1/2
[9]を外すと
x1=x2=x3=x4=x5=x6=x7=1/6
x8=5/6
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【3】E8~について(521)
(0,0,0,0,0,0,0,0)
(1,0,0,0,0,0,0,0)
(1,1/√3,0,0,0,0,0,0)
(1,1/√3,1/√6,0,0,0,0,0)
(1,1/√3,1/√6,1/√10,0,0,0,0)
(1,1/√3,1/√6,1/√10,1/√15,0,0,0)
(1,1/√3,1/√6,1/√10,1/√15,1/√12,0,0)
(1,1/√3,1/√6,1/√10,1/√15,1/√12,1/2,0)
(1,1/√3,1/√6,1/√10,1/√15,0,0,1/3)
>のすべての辺の長さが「SPLAG」第21章に一致することが確認された.
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