콕세터 군 H3

수학노트
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개요

  • 다음과 같이 정의되는 콕세터 군 $H_3$

$$ \left\langle r_1,r_2,r_3 \mid r_i^2=(r_3r_1)^2=(r_1r_2)^5=(r_2r_3)^3=1\right\rangle $$

  • 불변량

$$ \begin{array}{c|ccccc} & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{Coxeter} \\ \hline H_3 & 3 & 2,6,10 & 1,5,9 & 120 & 10 \end{array} $$

푸앵카레 다항식

  • $H_3$의 푸앵카레 다항식은 다음과 같다

$$ \begin{aligned} P_{W}(q)&=\sum_{w\in W}q^{\ell(w)} \\ &=1+3 q+5 q^2+7 q^3+9 q^4+11 q^5+12 q^6+12 q^7+12 q^8+12 q^9+11 q^{10}+9 q^{11}+7 q^{12}+5 q^{13}+3 q^{14}+q^{15} \end{aligned} $$


콕세터 원소

  • 콕세터 다항식, 즉 콕세터 원소의 특성다항식은 다음과 같다

$$ -(x+1) \left(x^2- \varphi x +1\right) $$ 여기서 $\varphi=\frac{1+\sqrt{5}}{2}$

  • 콕세터 다항식의 세 해는 $\zeta, \zeta^5,\zeta^9$로 주어지며 여기서 $\zeta=e^{2\pi i/10}$


루트 시스템

  • 30개의 원소로 구성
  • 다음과 같은 세 벡터가 simple system을 이룬다

$$ \begin{align} r_1= \beta(1+2 \alpha,1 , -2 \alpha) \\ r_2= \beta(-1-2 \alpha , 1 , 2 \alpha) \\ r_3= \beta(2 \alpha , -1-2 \alpha , 1) \end{align} $$ 여기서 $\alpha=\cos \pi/5, \beta=\cos 2\pi/5$

콕세터 군 H32.png

콕세터 평면으로의 사영

콕세터 군 H31.png


테이블

  • 원소

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

재미있는 사실

  • 2011년 9월 미국수학회보(Notices of the American Mathematical Society)의 표지에 콕세터 평면으로의 사영이 등장, 링크


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