"콕세터 군 H3"의 두 판 사이의 차이

2014년 6월 30일 (월) 06:28 판

개요

다음과 같이 정의되는 콕세터 군 $H_3$

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

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 평면으로의 사영

테이블

원소

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

매스매티카 파일 및 계산 리소스

https://docs.google.com/file/d/0B8XXo8Tve1cxQURSVzA3QXYzQTA/edit?usp=drivesdk

사전 형태의 자료

http://en.wikipedia.org/wiki/Binary_icosahedral_group