【第2章】対称群 \(S_3\)
【2-5】\(S_4\) の指標表と射影演算子
今対称群 \(S_4\) 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。| \(S_4\)の共役類 | \(C_{1}\) | \(C_{2}\) | \(C_{3}\) | \(C_{4}\) | \(C_{5}\) |
|---|---|---|---|---|---|
| 代表元の巡回表現 | \( e\) | \( (1,2) \) | \( (1,2,3) \) | \( (1,2,3,4) \) | \( (1,2)(3,4) \) |
| 元 | \(\sigma_1 \) | \( \{\sigma_{2},..,\sigma_{7}\} \) | \( \{\sigma_{8},..,\sigma_{15}\} \) | \( \{\sigma_{16},..,\sigma_{21}\} \) | \( \{\sigma_{22},\sigma_{23},\sigma_{24}\} \) |
| 恒等表現 \(\chi_{1}\) | \( 1\) | \( 1 \) | \(1 \) | \( 1 \) | \( 1 \) |
| 交代表現 \( \chi_{2}\) | \( 1\) | \(-1 \) | \( 1 \) | \( -1 \) | \( 1 \) |
| \( \chi_{3}\) | \( 2\) | \( 0 \) | \( -1 \) | \( 0 \) | \( 2 \) |
| \( \chi_{4}\) | \( 3\) | \( 1\) | \(0 \) | \( -1 \) | \( -1 \) |
| \( \chi_{5}\) | \( 3\) | \( -1 \) | \( 0 \) | \( 1 \) | \( -1 \) |
前節で \(S_4\) の左正則表現が求まったので、 \(S_4\) の射影演算子は(5.1)となります。
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 \(\{d_{1}=1,d_{2}=1,d_{3}=2,d_{4}=3,d_{5}=3\}\)となります。 また \(\vert G \vert\) は群の位数を表すので \(24\) となります。
\begin{align} P_{\rho}&=\frac{d_{\rho}}{\vert G \vert}\displaystyle \sum_{i=1}^{24} \chi_{\rho}(\sigma_i^{-1}) \cdot L_i \qquad [\rho=1,2,3,4,5] \\ \end{align}
\begin{align} &\qquad \Downarrow \notag \\ &\left\{ \begin{array}{l} P_1=\frac{1}{24} \biggl( L_1+ ( L_2+ ... +L_7)+( L_{8}+ ... +L_{15})+( L_{16}+ ... +L_{21})+( L_{22}+ ... +L_{24}) \biggr) \\ P_2=\frac{1}{24} \biggl( L_1 - ( L_2+ ... +L_7)+( L_{8}+ ... +L_{15})-( L_{16}+ ... +L_{21})+( L_{22}+ ... +L_{24}) \biggr) \\ P_3=\frac{2}{24} \biggl( 2L_1 -( L_{8}+ ... +L_{15})+2(L_{22}+... +L_{24})\biggr) \\ P_4=\frac{3}{24} \biggl( 3L_1+ ( L_2+ ... +L_7)-( L_{16} ... +L_{21})-( L_{22}+ ... +L_{24}) \biggr) \\ P_5=\frac{3}{24} \biggl( 3L_1- ( L_2+ ... +L_7)+( L_{16}+ ... +L_{21})-( L_{22}+ ... +L_{24}) \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以下の行列となります。
\begin{align} &P_1=\frac{1}{24}\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 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 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}{24}\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 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1\\ -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1\\ -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1\\ -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1\\ -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1\\ 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1\\ 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1\\ 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_3=\frac{1}{12}\begin{bmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2\\ 0 & 2 & -1 & -1 & 2 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 2 & -1 & 2 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 2 & -1 & -1 & -1 & 0 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 2 & -1 & -1 & 2 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 2 & -1 & 2 & 0 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 2 & -1 & -1 & -1 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & 2 & 2 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 2 & -1 & -1 & 2 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 2 & -1 & -1 & 2 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & 2 & 2 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & 2 & 2 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 2 & -1 & -1 & 2 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 2 & -1 & -1 & 2 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & 2 & 2 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ 0 & -1 & -1 & 2 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 2 & -1 & -1 & -1 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & -1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 2 & -1 & -1 & -1 & 0 & 0 & 0\\ 0 & 2 & -1 & -1 & 2 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 2 & -1 & 2 & 0 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 2 & -1 & -1 & 2 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 2 & -1 & 2 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2\end{bmatrix} \\ \notag \\ &P_4=\frac{1}{8}\begin{bmatrix}3 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1\\ 1 & 3 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 & 0 & 0 & 0 & -1 & 0 & -1 & 1 & -1 & -1\\ 1 & 0 & 3 & 0 & 0 & 0 & -1 & 1 & 1 & 1 & -1 & 1 & -1 & -1 & -1 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & -1 & 1\\ 1 & 0 & 0 & 3 & 0 & -1 & 0 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & -1 & 0 & 0 & 0 & -1 & 0 & -1 & 1 & -1\\ 1 & -1 & 0 & 0 & 3 & 0 & 0 & -1 & -1 & 1 & 1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & -1 & 1 & -1 & -1\\ 1 & 0 & 0 & -1 & 0 & 3 & 0 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & -1 & 1 & -1\\ 1 & 0 & -1 & 0 & 0 & 0 & 3 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & 1 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & -1 & 1\\ 0 & 1 & 1 & 1 & -1 & -1 & -1 & 3 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 1 & -1 & -1 & -1 & 0 & 3 & -1 & 0 & 0 & -1 & -1 & 0 & -1 & 1 & -1 & 1 & 1 & -1 & 0 & 0 & 0\\ 0 & -1 & 1 & -1 & 1 & 1 & -1 & 0 & -1 & 3 & 0 & 0 & -1 & -1 & 0 & 1 & 1 & -1 & -1 & -1 & 1 & 0 & 0 & 0\\ 0 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & 0 & 0 & 3 & -1 & 0 & 0 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 0 & 0 & 0\\ 0 & -1 & 1 & -1 & 1 & 1 & -1 & -1 & 0 & 0 & -1 & 3 & 0 & 0 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & 0 & 0 & 0\\ 0 & 1 & -1 & -1 & -1 & 1 & 1 & 0 & -1 & -1 & 0 & 0 & 3 & -1 & 0 & 1 & -1 & 1 & 1 & -1 & -1 & 0 & 0 & 0\\ 0 & -1 & -1 & 1 & 1 & -1 & 1 & 0 & -1 & -1 & 0 & 0 & -1 & 3 & 0 & -1 & -1 & 1 & -1 & 1 & 1 & 0 & 0 & 0\\ 0 & 1 & -1 & -1 & -1 & 1 & 1 & -1 & 0 & 0 & -1 & -1 & 0 & 0 & 3 & -1 & 1 & -1 & -1 & 1 & 1 & 0 & 0 & 0\\ -1 & 0 & 0 & -1 & 0 & -1 & 0 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 3 & 0 & 0 & 0 & -1 & 0 & 1 & -1 & 1\\ -1 & 0 & -1 & 0 & 0 & 0 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & 1 & 0 & 3 & -1 & 0 & 0 & 0 & 1 & 1 & -1\\ -1 & 0 & -1 & 0 & 0 & 0 & -1 & 1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & 0 & -1 & 3 & 0 & 0 & 0 & 1 & 1 & -1\\ -1 & -1 & 0 & 0 & -1 & 0 & 0 & -1 & 1 & -1 & 1 & 1 & 1 & -1 & -1 & 0 & 0 & 0 & 3 & 0 & -1 & -1 & 1 & 1\\ -1 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 0 & 0 & 0 & 3 & 0 & 1 & -1 & 1\\ -1 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & -1 & 1 & -1 & -1 & -1 & 1 & 1 & 0 & 0 & 0 & -1 & 0 & 3 & -1 & 1 & 1\\ -1 & 1 & -1 & -1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & -1 & 1 & -1 & 3 & -1 & -1\\ -1 & -1 & -1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & 1 & -1 & 1 & -1 & 3 & -1\\ -1 & -1 & 1 & -1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & 3\end{bmatrix} \\ \notag \\ &P_5=\frac{1}{8}\begin{bmatrix}3 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1\\ -1 & 3 & 0 & 0 & -1 & 0 & 0 & -1 & -1 & 1 & 1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & -1 & -1 & 1 & 1\\ -1 & 0 & 3 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & 1 & 0 & -1 & -1 & 0 & 0 & 0 & 1 & 1 & -1\\ -1 & 0 & 0 & 3 & 0 & -1 & 0 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & 1 & -1 & 1\\ -1 & -1 & 0 & 0 & 3 & 0 & 0 & 1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 & 0 & 0 & 0 & -1 & 0 & -1 & -1 & 1 & 1\\ -1 & 0 & 0 & -1 & 0 & 3 & 0 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & -1 & 0 & 0 & 0 & -1 & 0 & 1 & -1 & 1\\ -1 & 0 & -1 & 0 & 0 & 0 & 3 & 1 & 1 & 1 & -1 & 1 & -1 & -1 & -1 & 0 & -1 & -1 & 0 & 0 & 0 & 1 & 1 & -1\\ 0 & -1 & -1 & -1 & 1 & 1 & 1 & 3 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 0 & 0 & 0\\ 0 & -1 & -1 & -1 & 1 & 1 & 1 & 0 & 3 & -1 & 0 & 0 & -1 & -1 & 0 & 1 & -1 & 1 & -1 & -1 & 1 & 0 & 0 & 0\\ 0 & 1 & -1 & 1 & -1 & -1 & 1 & 0 & -1 & 3 & 0 & 0 & -1 & -1 & 0 & -1 & -1 & 1 & 1 & 1 & -1 & 0 & 0 & 0\\ 0 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 0 & 0 & 3 & -1 & 0 & 0 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & 0 & 0 & 0\\ 0 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 0 & 0 & -1 & 3 & 0 & 0 & -1 & 1 & 1 & -1 & -1 & -1 & 1 & 0 & 0 & 0\\ 0 & -1 & 1 & 1 & 1 & -1 & -1 & 0 & -1 & -1 & 0 & 0 & 3 & -1 & 0 & -1 & 1 & -1 & -1 & 1 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & -1 & -1 & 1 & -1 & 0 & -1 & -1 & 0 & 0 & -1 & 3 & 0 & 1 & 1 & -1 & 1 & -1 & -1 & 0 & 0 & 0\\ 0 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & 0 & 0 & -1 & -1 & 0 & 0 & 3 & 1 & -1 & 1 & 1 & -1 & -1 & 0 & 0 & 0\\ 1 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & 3 & 0 & 0 & 0 & -1 & 0 & -1 & 1 & -1\\ 1 & 0 & -1 & 0 & 0 & 0 & -1 & 1 & -1 & -1 & -1 & 1 & 1 & 1 & -1 & 0 & 3 & -1 & 0 & 0 & 0 & -1 & -1 & 1\\ 1 & 0 & -1 & 0 & 0 & 0 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & 1 & 0 & -1 & 3 & 0 & 0 & 0 & -1 & -1 & 1\\ 1 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & -1 & 1 & -1 & -1 & -1 & 1 & 1 & 0 & 0 & 0 & 3 & 0 & -1 & 1 & -1 & -1\\ 1 & 0 & 0 & -1 & 0 & -1 & 0 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & -1 & 0 & 0 & 0 & 3 & 0 & -1 & 1 & -1\\ 1 & -1 & 0 & 0 & -1 & 0 & 0 & -1 & 1 & -1 & 1 & 1 & 1 & -1 & -1 & 0 & 0 & 0 & -1 & 0 & 3 & 1 & -1 & -1\\ -1 & -1 & 1 & 1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 1 & -1 & 1 & 3 & -1 & -1\\ -1 & 1 & 1 & -1 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & -1 & 1 & -1 & -1 & 3 & -1\\ -1 & 1 & -1 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 3\end{bmatrix} \\ \notag \\ \end{align}
