"콕세터 군 H3"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
(같은 사용자의 중간 판 5개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
−
* 다음과 같이 정의되는 콕세터 군 $H_3$
+
* 다음과 같이 정의되는 콕세터 군 <math>H_3</math>
−
$$
+
:<math>
−
\left\langle r_1,r_2,r_3 \mid r_i^2=(r_3r_1)^2=(r_1r_2)^3=(r_2r_3)^5=1\right\rangle
+
\left\langle r_1,r_2,r_3 \mid r_i^2=(r_3r_1)^2=(r_1r_2)^5=(r_2r_3)^3=1\right\rangle
−
$$
+
</math>
 
* 불변량
 
* 불변량
−
$$
+
:<math>
 
\begin{array}{c|ccccc}
 
\begin{array}{c|ccccc}
 
   & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{Coxeter} \\
 
   & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{Coxeter} \\
11번째 줄: 11번째 줄:
 
  H_3 & 3 & 2,6,10 & 1,5,9 & 120 & 10
 
  H_3 & 3 & 2,6,10 & 1,5,9 & 120 & 10
 
\end{array}
 
\end{array}
−
$$
+
</math>
−
 
 
  
 
==푸앵카레 다항식==
 
==푸앵카레 다항식==
−
* $H_3$의 푸앵카레 다항식은 다음과 같다
+
* <math>H_3</math>의 푸앵카레 다항식은 다음과 같다
−
$$
+
:<math>
 
\begin{aligned}
 
\begin{aligned}
 
P_{W}(q)&=\sum_{w\in W}q^{\ell(w)} \\
 
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}
 
&=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}
 
\end{aligned}
−
$$
+
</math>
  
  
 
==콕세터 원소==
 
==콕세터 원소==
 
* 콕세터 다항식, 즉 콕세터 원소의 특성다항식은 다음과 같다
 
* 콕세터 다항식, 즉 콕세터 원소의 특성다항식은 다음과 같다
−
$$
+
:<math>
 
-(x+1) \left(x^2- \varphi x +1\right)
 
-(x+1) \left(x^2- \varphi x +1\right)
−
$$
+
</math>
−
여기서 $\varphi=\frac{1+\sqrt{5}}{2}$
+
여기서 <math>\varphi=\frac{1+\sqrt{5}}{2}</math>
−
* 콕세터 다항식의 세 해는 $\zeta, \zeta^5,\zeta^9$로 주어지며 여기서 $\zeta=e^{2\pi i/10}$
+
* 콕세터 다항식의 세 해는 <math>\zeta, \zeta^5,\zeta^9</math>로 주어지며 여기서 <math>\zeta=e^{2\pi i/10}</math>
  
  
36번째 줄: 35번째 줄:
 
* 30개의 원소로 구성
 
* 30개의 원소로 구성
 
* 다음과 같은 세 벡터가 simple system을 이룬다
 
* 다음과 같은 세 벡터가 simple system을 이룬다
−
$$
+
:<math>
 
\begin{align}
 
\begin{align}
 
r_1= \beta(1+2 \alpha,1 , -2 \alpha)  \\
 
r_1= \beta(1+2 \alpha,1 , -2 \alpha)  \\
42번째 줄: 41번째 줄:
 
r_3= \beta(2 \alpha  , -1-2 \alpha  , 1)
 
r_3= \beta(2 \alpha  , -1-2 \alpha  , 1)
 
\end{align}
 
\end{align}
−
$$
+
</math>
−
여기서 $\alpha=\cos \pi/5, \beta=\cos 2\pi/5$
+
여기서 <math>\alpha=\cos \pi/5, \beta=\cos 2\pi/5</math>
  
 
[[파일:콕세터 군 H32.png]]
 
[[파일:콕세터 군 H32.png]]
53번째 줄: 52번째 줄:
 
==테이블==
 
==테이블==
 
* 원소
 
* 원소
−
$$
+
:<math>
 
\begin{array}{ccc}
 
\begin{array}{ccc}
 
   & w & \ell(w) \\
 
   & w & \ell(w) \\
 
\hline
 
\hline
−
  1 & s() & 0 \\
+
  1 & \{\} & 0 \\
−
  2 & s(1) & 1 \\
+
  2 & \{1\} & 1 \\
−
  3 & s(2) & 1 \\
+
  3 & \{2\} & 1 \\
−
  4 & s(3) & 1 \\
+
  4 & \{3\} & 1 \\
−
  5 & s(2,1) & 2 \\
+
  5 & \{2,1\} & 2 \\
−
  6 & s(3,1) & 2 \\
+
  6 & \{3,1\} & 2 \\
−
  7 & s(1,2) & 2 \\
+
  7 & \{1,2\} & 2 \\
−
  8 & s(3,2) & 2 \\
+
  8 & \{3,2\} & 2 \\
−
  9 & s(2,3) & 2 \\
+
  9 & \{2,3\} & 2 \\
−
  10 & s(1,2,1) & 3 \\
+
  10 & \{1,2,1\} & 3 \\
−
  11 & s(3,2,1) & 3 \\
+
  11 & \{3,2,1\} & 3 \\
−
  12 & s(2,3,1) & 3 \\
+
  12 & \{2,3,1\} & 3 \\
−
  13 & s(2,1,2) & 3 \\
+
  13 & \{2,1,2\} & 3 \\
−
  14 & s(3,1,2) & 3 \\
+
  14 & \{3,1,2\} & 3 \\
−
  15 & s(2,3,2) & 3 \\
+
  15 & \{2,3,2\} & 3 \\
−
  16 & s(1,2,3) & 3 \\
+
  16 & \{1,2,3\} & 3 \\
−
  17 & s(2,1,2,1) & 4 \\
+
  17 & \{2,1,2,1\} & 4 \\
−
  18 & s(3,1,2,1) & 4 \\
+
  18 & \{3,1,2,1\} & 4 \\
−
  19 & s(2,3,2,1) & 4 \\
+
  19 & \{2,3,2,1\} & 4 \\
−
  20 & s(1,2,3,1) & 4 \\
+
  20 & \{1,2,3,1\} & 4 \\
−
  21 & s(1,2,1,2) & 4 \\
+
  21 & \{1,2,1,2\} & 4 \\
−
  22 & s(3,2,1,2) & 4 \\
+
  22 & \{3,2,1,2\} & 4 \\
−
  23 & s(2,3,1,2) & 4 \\
+
  23 & \{2,3,1,2\} & 4 \\
−
  24 & s(1,2,3,2) & 4 \\
+
  24 & \{1,2,3,2\} & 4 \\
−
  25 & s(2,1,2,3) & 4 \\
+
  25 & \{2,1,2,3\} & 4 \\
−
  26 & s(1,2,1,2,1) & 5 \\
+
  26 & \{1,2,1,2,1\} & 5 \\
−
  27 & s(3,2,1,2,1) & 5 \\
+
  27 & \{3,2,1,2,1\} & 5 \\
−
  28 & s(2,3,1,2,1) & 5 \\
+
  28 & \{2,3,1,2,1\} & 5 \\
−
  29 & s(1,2,3,2,1) & 5 \\
+
  29 & \{1,2,3,2,1\} & 5 \\
−
  30 & s(2,1,2,3,1) & 5 \\
+
  30 & \{2,1,2,3,1\} & 5 \\
−
  31 & s(3,1,2,1,2) & 5 \\
+
  31 & \{3,1,2,1,2\} & 5 \\
−
  32 & s(2,3,2,1,2) & 5 \\
+
  32 & \{2,3,2,1,2\} & 5 \\
−
  33 & s(1,2,3,1,2) & 5 \\
+
  33 & \{1,2,3,1,2\} & 5 \\
−
  34 & s(2,1,2,3,2) & 5 \\
+
  34 & \{2,1,2,3,2\} & 5 \\
−
  35 & s(1,2,1,2,3) & 5 \\
+
  35 & \{1,2,1,2,3\} & 5 \\
−
  36 & s(3,2,1,2,3) & 5 \\
+
  36 & \{3,2,1,2,3\} & 5 \\
−
  37 & s(3,1,2,1,2,1) & 6 \\
+
  37 & \{3,1,2,1,2,1\} & 6 \\
−
  38 & s(2,3,2,1,2,1) & 6 \\
+
  38 & \{2,3,2,1,2,1\} & 6 \\
−
  39 & s(1,2,3,1,2,1) & 6 \\
+
  39 & \{1,2,3,1,2,1\} & 6 \\
−
  40 & s(2,1,2,3,2,1) & 6 \\
+
  40 & \{2,1,2,3,2,1\} & 6 \\
−
  41 & s(1,2,1,2,3,1) & 6 \\
+
  41 & \{1,2,1,2,3,1\} & 6 \\
−
  42 & s(3,2,1,2,3,1) & 6 \\
+
  42 & \{3,2,1,2,3,1\} & 6 \\
−
  43 & s(2,3,1,2,1,2) & 6 \\
+
  43 & \{2,3,1,2,1,2\} & 6 \\
−
  44 & s(1,2,3,2,1,2) & 6 \\
+
  44 & \{1,2,3,2,1,2\} & 6 \\
−
  45 & s(2,1,2,3,1,2) & 6 \\
+
  45 & \{2,1,2,3,1,2\} & 6 \\
−
  46 & s(1,2,1,2,3,2) & 6 \\
+
  46 & \{1,2,1,2,3,2\} & 6 \\
−
  47 & s(3,2,1,2,3,2) & 6 \\
+
  47 & \{3,2,1,2,3,2\} & 6 \\
−
  48 & s(3,1,2,1,2,3) & 6 \\
+
  48 & \{3,1,2,1,2,3\} & 6 \\
−
  49 & s(2,3,1,2,1,2,1) & 7 \\
+
  49 & \{2,3,1,2,1,2,1\} & 7 \\
−
  50 & s(1,2,3,2,1,2,1) & 7 \\
+
  50 & \{1,2,3,2,1,2,1\} & 7 \\
−
  51 & s(2,1,2,3,1,2,1) & 7 \\
+
  51 & \{2,1,2,3,1,2,1\} & 7 \\
−
  52 & s(1,2,1,2,3,2,1) & 7 \\
+
  52 & \{1,2,1,2,3,2,1\} & 7 \\
−
  53 & s(3,2,1,2,3,2,1) & 7 \\
+
  53 & \{3,2,1,2,3,2,1\} & 7 \\
−
  54 & s(3,1,2,1,2,3,1) & 7 \\
+
  54 & \{3,1,2,1,2,3,1\} & 7 \\
−
  55 & s(1,2,3,1,2,1,2) & 7 \\
+
  55 & \{1,2,3,1,2,1,2\} & 7 \\
−
  56 & s(2,1,2,3,2,1,2) & 7 \\
+
  56 & \{2,1,2,3,2,1,2\} & 7 \\
−
  57 & s(1,2,1,2,3,1,2) & 7 \\
+
  57 & \{1,2,1,2,3,1,2\} & 7 \\
−
  58 & s(3,2,1,2,3,1,2) & 7 \\
+
  58 & \{3,2,1,2,3,1,2\} & 7 \\
−
  59 & s(3,1,2,1,2,3,2) & 7 \\
+
  59 & \{3,1,2,1,2,3,2\} & 7 \\
−
  60 & s(2,3,1,2,1,2,3) & 7 \\
+
  60 & \{2,3,1,2,1,2,3\} & 7 \\
−
  61 & s(1,2,3,1,2,1,2,1) & 8 \\
+
  61 & \{1,2,3,1,2,1,2,1\} & 8 \\
−
  62 & s(2,1,2,3,2,1,2,1) & 8 \\
+
  62 & \{2,1,2,3,2,1,2,1\} & 8 \\
−
  63 & s(1,2,1,2,3,1,2,1) & 8 \\
+
  63 & \{1,2,1,2,3,1,2,1\} & 8 \\
−
  64 & s(3,2,1,2,3,1,2,1) & 8 \\
+
  64 & \{3,2,1,2,3,1,2,1\} & 8 \\
−
  65 & s(3,1,2,1,2,3,2,1) & 8 \\
+
  65 & \{3,1,2,1,2,3,2,1\} & 8 \\
−
  66 & s(2,3,1,2,1,2,3,1) & 8 \\
+
  66 & \{2,3,1,2,1,2,3,1\} & 8 \\
−
  67 & s(2,1,2,3,1,2,1,2) & 8 \\
+
  67 & \{2,1,2,3,1,2,1,2\} & 8 \\
−
  68 & s(1,2,1,2,3,2,1,2) & 8 \\
+
  68 & \{1,2,1,2,3,2,1,2\} & 8 \\
−
  69 & s(3,2,1,2,3,2,1,2) & 8 \\
+
  69 & \{3,2,1,2,3,2,1,2\} & 8 \\
−
  70 & s(3,1,2,1,2,3,1,2) & 8 \\
+
  70 & \{3,1,2,1,2,3,1,2\} & 8 \\
−
  71 & s(2,3,1,2,1,2,3,2) & 8 \\
+
  71 & \{2,3,1,2,1,2,3,2\} & 8 \\
−
  72 & s(1,2,3,1,2,1,2,3) & 8 \\
+
  72 & \{1,2,3,1,2,1,2,3\} & 8 \\
−
  73 & s(2,1,2,3,1,2,1,2,1) & 9 \\
+
  73 & \{2,1,2,3,1,2,1,2,1\} & 9 \\
−
  74 & s(1,2,1,2,3,2,1,2,1) & 9 \\
+
  74 & \{1,2,1,2,3,2,1,2,1\} & 9 \\
−
  75 & s(3,2,1,2,3,2,1,2,1) & 9 \\
+
  75 & \{3,2,1,2,3,2,1,2,1\} & 9 \\
−
  76 & s(3,1,2,1,2,3,1,2,1) & 9 \\
+
  76 & \{3,1,2,1,2,3,1,2,1\} & 9 \\
−
  77 & s(2,3,1,2,1,2,3,2,1) & 9 \\
+
  77 & \{2,3,1,2,1,2,3,2,1\} & 9 \\
−
  78 & s(1,2,3,1,2,1,2,3,1) & 9 \\
+
  78 & \{1,2,3,1,2,1,2,3,1\} & 9 \\
−
  79 & s(1,2,1,2,3,1,2,1,2) & 9 \\
+
  79 & \{1,2,1,2,3,1,2,1,2\} & 9 \\
−
  80 & s(3,2,1,2,3,1,2,1,2) & 9 \\
+
  80 & \{3,2,1,2,3,1,2,1,2\} & 9 \\
−
  81 & s(3,1,2,1,2,3,2,1,2) & 9 \\
+
  81 & \{3,1,2,1,2,3,2,1,2\} & 9 \\
−
  82 & s(2,3,1,2,1,2,3,1,2) & 9 \\
+
  82 & \{2,3,1,2,1,2,3,1,2\} & 9 \\
−
  83 & s(1,2,3,1,2,1,2,3,2) & 9 \\
+
  83 & \{1,2,3,1,2,1,2,3,2\} & 9 \\
−
  84 & s(2,1,2,3,1,2,1,2,3) & 9 \\
+
  84 & \{2,1,2,3,1,2,1,2,3\} & 9 \\
−
  85 & s(1,2,1,2,3,1,2,1,2,1) & 10 \\
+
  85 & \{1,2,1,2,3,1,2,1,2,1\} & 10 \\
−
  86 & s(3,2,1,2,3,1,2,1,2,1) & 10 \\
+
  86 & \{3,2,1,2,3,1,2,1,2,1\} & 10 \\
−
  87 & s(3,1,2,1,2,3,2,1,2,1) & 10 \\
+
  87 & \{3,1,2,1,2,3,2,1,2,1\} & 10 \\
−
  88 & s(2,3,1,2,1,2,3,1,2,1) & 10 \\
+
  88 & \{2,3,1,2,1,2,3,1,2,1\} & 10 \\
−
  89 & s(1,2,3,1,2,1,2,3,2,1) & 10 \\
+
  89 & \{1,2,3,1,2,1,2,3,2,1\} & 10 \\
−
  90 & s(2,1,2,3,1,2,1,2,3,1) & 10 \\
+
  90 & \{2,1,2,3,1,2,1,2,3,1\} & 10 \\
−
  91 & s(3,1,2,1,2,3,1,2,1,2) & 10 \\
+
  91 & \{3,1,2,1,2,3,1,2,1,2\} & 10 \\
−
  92 & s(2,3,1,2,1,2,3,2,1,2) & 10 \\
+
  92 & \{2,3,1,2,1,2,3,2,1,2\} & 10 \\
−
  93 & s(1,2,3,1,2,1,2,3,1,2) & 10 \\
+
  93 & \{1,2,3,1,2,1,2,3,1,2\} & 10 \\
−
  94 & s(2,1,2,3,1,2,1,2,3,2) & 10 \\
+
  94 & \{2,1,2,3,1,2,1,2,3,2\} & 10 \\
−
  95 & s(3,2,1,2,3,1,2,1,2,3) & 10 \\
+
  95 & \{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 \\
+
  96 & \{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 \\
+
  97 & \{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 \\
+
  98 & \{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 \\
+
  99 & \{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 \\
+
  100 & \{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 \\
+
  101 & \{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 \\
+
  102 & \{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 \\
+
  103 & \{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 \\
+
  104 & \{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 \\
+
  105 & \{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 \\
+
  106 & \{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 \\
+
  107 & \{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 \\
+
  108 & \{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 \\
+
  109 & \{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 \\
+
  110 & \{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 \\
+
  111 & \{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 \\
+
  112 & \{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 \\
+
  113 & \{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 \\
+
  114 & \{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 \\
+
  115 & \{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 \\
+
  116 & \{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 \\
+
  117 & \{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 \\
+
  118 & \{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 \\
+
  119 & \{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
+
  120 & \{3,2,1,2,3,1,2,1,2,3,1,2,1,2,1\} & 15
 
\end{array}
 
\end{array}
−
$$
+
</math>
  
 
==재미있는 사실==
 
==재미있는 사실==
187번째 줄: 186번째 줄:
 
* [[교대군 A5]]
 
* [[교대군 A5]]
 
* [[정이십면체 뫼비우스 변환군]]
 
* [[정이십면체 뫼비우스 변환군]]
 +
* [[콕세터 군 H4]]
  
  
196번째 줄: 196번째 줄:
 
* http://en.wikipedia.org/wiki/Binary_icosahedral_group
 
* http://en.wikipedia.org/wiki/Binary_icosahedral_group
 
[[분류:리군과 리대수]]
 
[[분류:리군과 리대수]]
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q4913898 Q4913898]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'binary'}, {'LOWER': 'icosahedral'}, {'LEMMA': 'group'}]

2021년 2월 17일 (수) 02:22 기준 최신판

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


관련된 항목들


매스매티카 파일 및 계산 리소스


사전 형태의 자료

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'binary'}, {'LOWER': 'icosahedral'}, {'LEMMA': 'group'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=콕세터_군_H3&oldid=51993"