"정이십면체 뫼비우스 변환군"의 두 판 사이의 차이

2012년 7월 24일 (화) 17:37 판

\(G_{60}=\langle S,T|S^5=T^2=(TS)^3=1\rangle\subset \operatorname{PSL}(2,\mathbb{C})\)

\(S=\left( \begin{array}{cc} \zeta ^3 & 0 \\ 0 & \zeta ^2 \end{array} \right)\) order 5

\(\sqrt{5}T=\left( \begin{array}{cc} \zeta -\zeta ^4 & \zeta ^3-\zeta ^2 \\ \zeta ^3-\zeta ^2 & \zeta ^4-\zeta \end{array} \right)\) order 2

\(W=TS\) : order 3

vertex points
- \(V=F_1=z_1z_2(z_1^{10}+11z_1^5z_2^5-z_2^{10})\)
face points
- \(F=F_2=-(z_1^{20}+z_2^{20})+228(z_1^{15}z_2^{5}-z_1^{5}z_2^{15})-494z_1^{10}z_2^{10}\)
edge points
- \(E=F_3=(z_1^{30}+z_2^{30})+522(z_1^{25}z_2^{5}-z_1^{5}z_2^{25})-10005(z_1^{20}z_2^{10}+z_1^{10}z_2^{20})\)
syzygy relation
\(1728F_1^5-F_2^3-F_3^2=0\) 또는 \(1728V^5-E^2-F^3=0\)
\(F_2=HF_1\)
\(F_3=JF_1\)

@@ 22번째 줄: / 22번째 줄: @@
 <math>W=TS</math> : order 3
-* 정이십면체의 대칭은 교대군 <math>A_5</math>
-*  생성원<br><math>S=\begin{pmatrix} \zeta_{10} & 0 \\ 0 & \zeta_{10} \end{pmatrix} </math>, <math>T={\begin{pmatrix} -1 & g \\ g & 1 \end{pmatrix}}</math>,  <math>g=\frac{\sqrt{5}-1}{2}</math><br>
@@ 45번째 줄: / 36번째 줄: @@
 ** <math>E=F_3=(z_1^{30}+z_2^{30})+522(z_1^{25}z_2^{5}-z_1^{5}z_2^{25})-10005(z_1^{20}z_2^{10}+z_1^{10}z_2^{20})</math>
 *  syzygy relation<br><math>1728F_1^5-F_2^3-F_3^2=0</math> 또는 <math>1728V^5-E^2-F^3=0</math><br>
+* <math>F_2=HF_1</math>
+* <math>F_3=JF_1</math>