【第4章】Frobenius群 \(F_{20}\)
【4-5】\(F_{20}\) の指標表と射影演算子
今 \(F_{20}\) の指標表が下記【表4】が既知とすることから話を始めます。| \(F_{20}\)の共役類 | \(C_{1}\) | \(C_{2}\) | \(C_{3}\) | \(C_{4}\) | \(C_{5}\) |
|---|---|---|---|---|---|
| 代表元の巡回置換 | \( e\) | \( (1,2,3,4,5) \) | \( (2,3,4,5) \) | \( (2,5)(3,4) \) | \( (2,4,5,3) \) |
| \(F_{20}\) の元 | \(\sigma_1 \) | \( \{\sigma_{2},..,\sigma_{5}\} \) | \( \{\sigma_{6},..,\sigma_{10}\} \) | \( \{\sigma_{11},..,\sigma_{15}\} \) | \( \{\sigma_{16},..,\sigma_{20}\} \) |
| 共役類の元の数 | \( 1\) | \( 4 \) | \( 5 \) | \( 5 \) | \( 5 \) |
| \(\chi_{1}\) | \( 1\) | \( 1 \) | \(1 \) | \( 1 \) | \( 1 \) |
| \( \chi_{2}\) | \( 1\) | \( 1\) | \(\mathit{i} \) | \( -1 \) | \( -\mathit{i} \) |
| \( \chi_{3}\) | \( 1\) | \(1 \) | \( -1 \) | \( 1 \) | \( -1 \) |
| \( \chi_{4}\) | \( 1\) | \( 1 \) | \( -\mathit{i} \) | \( -1 \) | \( \mathit{i} \) |
| \( \chi_{5}\) | \( 4\) | \(-1 \) | \( 0 \) | \( 0 \) | \( 0 \) |
前節で \(F_{20}\) の左正則表現が求まったので、 \(F_{20}\) の射影演算子は(5.1)(5.2)と記述できます。
ここで、\(d_{\rho}\) は表現の次元数を表すので、共役類\(C_1\) の指標と同じ \(\{d_{1}=1,d_{2}=1,d_{3}=1,d_{4}=1,d_{5}=4\}\)となります。 また \(\vert G \vert\) は群の位数を表すので \(20\) となります。
\begin{align} P_{\rho}&=\frac{d_{\rho}}{\vert G \vert}\displaystyle \sum_{i=1}^{20} \chi_{\rho} \bbox[#FFFF00]{(\sigma_i^{-1})} \cdot L_i \qquad [\rho=1,2,3,4,5] \\ \notag \\ &\qquad \Downarrow \notag \\ \end{align}
\begin{align} \notag \\ &\left\{ \begin{array}{l} P_1=\frac{1}{20}\biggl( L_1+\bigl(L_2+L_3+L_4+L_5\bigr)+ \bigl(L_6+...+L_{10}\bigr)+\bigl(L_{11}+...+L_{15}\bigr)+\bigl(L_{16}+...+L_{20}\bigr) \biggr) \\ P_2=\frac{1}{20}\biggl( L_1+\bigl(L_2+L_3+L_4+L_5\bigr) \bbox[#FFFF00]{-i}\bigl(L_6+...+L_{10}\bigr)-\bigl(L_{11}+...+L_{15}\bigr) \bbox[#FFFF00]{+i}\bigl(L_{16}+...+L_{20}\bigr) \biggr) \\ P_3=\frac{1}{20}\biggl( L_1+\bigl(L_2+L_3+L_4+L_5\bigr)-\bigl(L_6+...+L_{10}\bigr)+\bigl(L_{11}+...+L_{15}\bigr)-\bigl(L_{16}+...+L_{20}\bigr) \biggr) \\ P_4=\frac{1}{20}\biggl( L_1+\bigl(L_2+L_3+L_4+L_5\bigr) \bbox[#FFFF00]{+i}\bigl(L_6+...+L_{10}\bigr)-\bigl(L_{11}+...+L_{15}\bigr) \bbox[#FFFF00]{-i}\bigl(L_{16}+...+L_{20}\bigr) \biggr) \\ P_5=\frac{4}{20}\biggl( 4L_1-\bigl(L_2+L_3+L_4+L_5\bigr) \biggr) \\ \end{array} \right.\\ \end{align}
具体的に左正則表現の行列の値を代入すると射影演算子の行列表現は(5.3)～(5.8)となります。
\begin{align} &P_1=\frac{1}{20}\begin{bmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\end{bmatrix} \\ \notag \\ &P_{2}=\frac{1}{20}\begin{bmatrix} 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i\\ 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i\\ 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i\\ 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i\\ 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i\\ -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1\\ -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1\\ -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1\\ -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1\\ -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i & -1 & -1 & -1 & -1 & -1\\ -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i\\ -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i\\ -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i\\ -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i\\ -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1 & i & i & i & i & i\\ i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1\\ i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1\\ i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1\\ i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1\\ i & i & i & i & i & -1 & -1 & -1 & -1 & -1 & -i & -i & -i & -i & -i & 1 & 1 & 1 & 1 & 1\end{bmatrix} \\ \notag \\ &P_{3}=\frac{1}{20}\begin{bmatrix}1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\end{bmatrix} \\ \notag \\ &P_{4}=\frac{1}{20}\begin{bmatrix}1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i\\ 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i\\ 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i\\ 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i\\ 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i\\ i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1\\ i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1\\ i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1\\ i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1\\ i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1\\ -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i\\ -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i\\ -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i\\ -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i\\ -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1 & -i & -i & -i & -i & -i\\ -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1\\ -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1\\ -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1\\ -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1\\ -i & -i & -i & -i & -i & -1 & -1 & -1 & -1 & -1 & i & i & i & i & i & 1 & 1 & 1 & 1 & 1\end{bmatrix} \\ \\ \notag \\ &P_{5}=\frac{4}{20}\begin{bmatrix}4 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 4 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & -1 & 4 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & -1 & -1 & 4 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & -1 & -1 & -1 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 4 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 4 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & -1 & 4 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 4 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 4 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 4 & -1 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 4 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & 4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & -1 & -1 & -1 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 4 & -1 & -1 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 4 & -1 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 4 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & 4\end{bmatrix} \\ \notag \\ \end{align}
