【第2章】対称群 \(S_4\)
【2-1】 \(S_4\) の既約分解の流れ
下図Fig.1は、対称群 \(S_4\) の正則表現 \(L_i\) を既約表現 \(\widetilde{L_i}\) に分解する計算の流れを数に示します。
(STEP1)
\(L_i\)：\(S_4\) の元の（左）正則表現行列。\([i=1,2,..,24]\)
\(Q_i\)：指標表から計算される射影演算子の固有ベクトルより生成されるブロック分解の為の変換行列。
因みにこの行列 \(Q\) で変換された行列 \(\breve{L_{i}}\) は既約表現ではありません。既約表現の途中段階だと思ってください。
(STEP2)
\(R_{2}\)：(4x4)の小ブロック \(B_i\) を(2x2)+(2x2)の既約表現 \(\widetilde{B_i}\) にまで既約分解する変換行列。
\(R_{3}\)：(9x9)の小ブロック \(D_i\) を(3x3)+(3x3+(3x3))の既約表現 \(\widetilde{D_i}\) にまで既約分解する変換行列。
\(R_{4}\)：(9x9)の小ブロック \(F_i\) を(3x3)+(3x3)+(3x3)の既約表現 \(\widetilde{F_i}\) にまで既約分解する変換行列。
(STEP3)
\(T\)：行列 \(Q\) と \(\{R_{2},R_{3},R_{4}\}\) を合体して正則表現 \(L_i\) を一気に既約表現 \(\widetilde{L_i}\) にまで分解する変換行列。
本章の最終目標はこの正則表現を完全に既約表現に変換する行列 \(T\) を求める計算法を解説する事です。
【2-2】\(S_4\) の元と番号付け
先ずは対称群 \(S_4\) の元それぞれに対応する（左）正則表現を求める為の準備から始めます。表現論では、群の元の順番付けが非常に重要となります。
何故ならば元の番号が変わると対応する元の行列表示が全く異なってしまうからです。下記【表1】では、共役類ごと元がまとまるように並べられ番号付けがされております。
| \(S_4\) の元 | \(\sigma_{1}\) | \(\sigma_{2}\) | \(\sigma_{3}\) | \(\sigma_{4}\) | \(\sigma_{5}\) | \(\sigma_{6}\) |
|---|---|---|---|---|---|---|
| 置換2行表示 | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 2 & 3 & 4\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 2 & 4 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 3 & 2 & 4\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 4 & 3 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 1 & 3 & 4\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 2 & 1 & 4\end{pmatrix} \) |
| 巡回表現 | \( e\) | \( (3,4) \) | \( (2,3) \) | \( (2,4) \) | \( (1,2) \) | \( (1,3) \) |
| \(S_4\) の元 | \(\sigma_{7}\) | \(\sigma_{8}\) | \(\sigma_{9}\) | \(\sigma_{10}\) | \(\sigma_{11}\) | \(\sigma_{12}\) |
|---|---|---|---|---|---|---|
| 置換2行表示 | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 2 & 3 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 3 & 4 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 4 & 2 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 3 & 1 & 4\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 4 & 3 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 1 & 2 & 4\end{pmatrix} \) |
| 巡回表現 | \( (1,4) \) | \( (2,3,4) \) | \( (2,4,3) \) | \( (1,2,3) \) | \( (1,2,4) \) | \( (1,3,2) \) |
| \(S_4\) の元 | \(\sigma_{13}\) | \(\sigma_{14}\) | \(\sigma_{15}\) | \(\sigma_{16}\) | \(\sigma_{17}\) | \(\sigma_{18}\) |
|---|---|---|---|---|---|---|
| 置換2行表示 | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 2 & 4 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 1 & 3 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 2 & 1 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 3 & 4 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 4 & 1 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 1 & 4 & 2\end{pmatrix} \) |
| 巡回表現 | \( (1,3,4) \) | \( (1,4,2) \) | \( (1,4,3) \) | \( (1,2,3,4) \) | \( (1,2,4,3) \) | \( (1,3,4,2) \) |
| \(S_4\) の元 | \(\sigma_{19}\) | \(\sigma_{20}\) | \(\sigma_{21}\) | \(\sigma_{22}\) | \(\sigma_{23}\) | \(\sigma_{24}\) |
|---|---|---|---|---|---|---|
| 置換2行表示 | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 4 & 2 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 1 & 2 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 3 & 1 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 1 & 4 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 4 & 1 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 3 & 2 & 1\end{pmatrix} \) |
| 巡回表現 | \( (1,3,2,4) \) | \( (1,4,3,2) \) | \( (1,4,2,3) \) | \( (1,2)(3,4) \) | \( (1,3)(2,4) \) | \( (1,4)(2,3) \) |
【2-3】\(S_4\) の積表
対称群 \(S_4\) の元同士は積の演算が定義されています。例えば(3.1)では \( \sigma_2 \circ \sigma_5\) の積の例を示してあります。 注意する事は、積 \( \sigma_2 \circ \sigma_5\) は、 \( (1,2,3,4) \) という数字の順列に対して、右の \((\sigma_5)\) から左 \((\sigma_2)\) に 数字の置換がなされてゆくという順番です。
全ての元同士の積の演算結果を【表2】にまとめておきました。
\begin{align} &[e.g.] \qquad \sigma_2 \circ \sigma_5=\begin{pmatrix} 1&2&3&4 \\ 1& 2&4&3 \end{pmatrix} \begin{pmatrix} 1&2&3&4 \\ 2&1&3&4 \end{pmatrix} =\begin{pmatrix} 1&2&3&4 \\ 2&1&4&3 \end{pmatrix} =\sigma_{22} \\ \end{align}
| \(i \backslash j\) | \(\sigma_1\) | \(\sigma_2\) | \(\sigma_3\) | \(\sigma_4\) | \(\sigma_5\) | \(\sigma_6\) | \(\sigma_7\) | \(\sigma_8\) | \(\sigma_9\) | \(\sigma_{10}\) | \(\sigma_{11}\) | \(\sigma_{12}\) | \(\sigma_{13}\) | \(\sigma_{14}\) | \(\sigma_{15}\) | \(\sigma_{16}\) | \(\sigma_{17}\) | \(\sigma_{18}\) | \(\sigma_{19}\) | \(\sigma_{20}\) | \(\sigma_{21}\) | \(\sigma_{22}\) | \(\sigma_{23}\) | \(\sigma_{24}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\sigma_{1}\) | \(\sigma_1\) | \(\sigma_2\) | \(\sigma_3\) | \(\sigma_4\) | \(\sigma_5\) | \(\sigma_6\) | \(\sigma_7\) | \(\sigma_8\) | \(\sigma_9\) | \(\sigma_{10}\) | \(\sigma_{11}\) | \(\sigma_{12}\) | \(\sigma_{13}\) | \(\sigma_{14}\) | \(\sigma_{15}\) | \(\sigma_{16}\) | \(\sigma_{17}\) | \(\sigma_{18}\) | \(\sigma_{19}\) | \(\sigma_{20}\) | \(\sigma_{21}\) | \(\sigma_{22}\) | \(\sigma_{23}\) | \(\sigma_{24}\) |
| \(\sigma_{2}\) | \(\sigma_2\) | \(\sigma_1\) | \(\sigma_9\) | \(\sigma_8\) | \(\sigma_{22}\) | \(\sigma_{15}\) | \(\sigma_{13}\) | \(\sigma_4\) | \(\sigma_3\) | \(\sigma_{17}\) | \(\sigma_{16}\) | \(\sigma_{20}\) | \(\sigma_7\) | \(\sigma_{18}\) | \(\sigma_6\) | \(\sigma_{11}\) | \(\sigma_{10}\) | \(\sigma_{14}\) | \(\sigma_{24}\) | \(\sigma_{12}\) | \(\sigma_{23}\) | \(\sigma_5\) | \(\sigma_{21}\) | \(\sigma_{19}\) |
| \(\sigma_{3}\) | \(\sigma_3\) | \(\sigma_8\) | \(\sigma_1\) | \(\sigma_9\) | \(\sigma_{12}\) | \(\sigma_{10}\) | \(\sigma_{24}\) | \(\sigma_2\) | \(\sigma_4\) | \(\sigma_6\) | \(\sigma_{19}\) | \(\sigma_5\) | \(\sigma_{16}\) | \(\sigma_{20}\) | \(\sigma_{21}\) | \(\sigma_{13}\) | \(\sigma_{23}\) | \(\sigma_{22}\) | \(\sigma_{11}\) | \(\sigma_{14}\) | \(\sigma_{15}\) | \(\sigma_{18}\) | \(\sigma_{17}\) | \(\sigma_7\) |
| \(\sigma_{4}\) | \(\sigma_4\) | \(\sigma_9\) | \(\sigma_8\) | \(\sigma_1\) | \(\sigma_{14}\) | \(\sigma_{23}\) | \(\sigma_{11}\) | \(\sigma_3\) | \(\sigma_2\) | \(\sigma_{21}\) | \(\sigma_7\) | \(\sigma_{18}\) | \(\sigma_{19}\) | \(\sigma_5\) | \(\sigma_{17}\) | \(\sigma_{24}\) | \(\sigma_{15}\) | \(\sigma_{12}\) | \(\sigma_{13}\) | \(\sigma_{22}\) | \(\sigma_{10}\) | \(\sigma_{20}\) | \(\sigma_6\) | \(\sigma_{16}\) |
| \(\sigma_{5}\) | \(\sigma_5\) | \(\sigma_{22}\) | \(\sigma_{10}\) | \(\sigma_{11}\) | \(\sigma_1\) | \(\sigma_{12}\) | \(\sigma_{14}\) | \(\sigma_{16}\) | \(\sigma_{17}\) | \(\sigma_3\) | \(\sigma_4\) | \(\sigma_6\) | \(\sigma_{18}\) | \(\sigma_7\) | \(\sigma_{20}\) | \(\sigma_8\) | \(\sigma_9\) | \(\sigma_{13}\) | \(\sigma_{23}\) | \(\sigma_{15}\) | \(\sigma_{24}\) | \(\sigma_2\) | \(\sigma_{19}\) | \(\sigma_{21}\) |
| \(\sigma_{6}\) | \(\sigma_6\) | \(\sigma_{13}\) | \(\sigma_{12}\) | \(\sigma_{23}\) | \(\sigma_{10}\) | \(\sigma_1\) | \(\sigma_{15}\) | \(\sigma_{18}\) | \(\sigma_{19}\) | \(\sigma_5\) | \(\sigma_{17}\) | \(\sigma_3\) | \(\sigma_2\) | \(\sigma_{21}\) | \(\sigma_7\) | \(\sigma_{22}\) | \(\sigma_{11}\) | \(\sigma_8\) | \(\sigma_9\) | \(\sigma_{24}\) | \(\sigma_{14}\) | \(\sigma_{16}\) | \(\sigma_4\) | \(\sigma_{20}\) |
| \(\sigma_{7}\) | \(\sigma_7\) | \(\sigma_{15}\) | \(\sigma_{24}\) | \(\sigma_{14}\) | \(\sigma_{11}\) | \(\sigma_{13}\) | \(\sigma_1\) | \(\sigma_{21}\) | \(\sigma_{20}\) | \(\sigma_{16}\) | \(\sigma_5\) | \(\sigma_{19}\) | \(\sigma_6\) | \(\sigma_4\) | \(\sigma_2\) | \(\sigma_{10}\) | \(\sigma_{22}\) | \(\sigma_{23}\) | \(\sigma_{12}\) | \(\sigma_9\) | \(\sigma_8\) | \(\sigma_{17}\) | \(\sigma_{18}\) | \(\sigma_3\) |
| \(\sigma_{8}\) | \(\sigma_8\) | \(\sigma_3\) | \(\sigma_4\) | \(\sigma_2\) | \(\sigma_{18}\) | \(\sigma_{21}\) | \(\sigma_{16}\) | \(\sigma_9\) | \(\sigma_1\) | \(\sigma_{23}\) | \(\sigma_{13}\) | \(\sigma_{14}\) | \(\sigma_{24}\) | \(\sigma_{22}\) | \(\sigma_{10}\) | \(\sigma_{19}\) | \(\sigma_6\) | \(\sigma_{20}\) | \(\sigma_7\) | \(\sigma_5\) | \(\sigma_{17}\) | \(\sigma_{12}\) | \(\sigma_{15}\) | \(\sigma_{11}\) |
| \(\sigma_{9}\) | \(\sigma_9\) | \(\sigma_4\) | \(\sigma_2\) | \(\sigma_3\) | \(\sigma_{20}\) | \(\sigma_{17}\) | \(\sigma_{19}\) | \(\sigma_1\) | \(\sigma_8\) | \(\sigma_{15}\) | \(\sigma_{24}\) | \(\sigma_{22}\) | \(\sigma_{11}\) | \(\sigma_{12}\) | \(\sigma_{23}\) | \(\sigma_7\) | \(\sigma_{21}\) | \(\sigma_5\) | \(\sigma_{16}\) | \(\sigma_{18}\) | \(\sigma_6\) | \(\sigma_{14}\) | \(\sigma_{10}\) | \(\sigma_{13}\) |
| \(\sigma_{10}\) | \(\sigma_{10}\) | \(\sigma_{16}\) | \(\sigma_5\) | \(\sigma_{17}\) | \(\sigma_6\) | \(\sigma_3\) | \(\sigma_{21}\) | \(\sigma_{22}\) | \(\sigma_{11}\) | \(\sigma_{12}\) | \(\sigma_{23}\) | \(\sigma_1\) | \(\sigma_8\) | \(\sigma_{15}\) | \(\sigma_{24}\) | \(\sigma_{18}\) | \(\sigma_{19}\) | \(\sigma_2\) | \(\sigma_4\) | \(\sigma_7\) | \(\sigma_{20}\) | \(\sigma_{13}\) | \(\sigma_9\) | \(\sigma_{14}\) |
| \(\sigma_{11}\) | \(\sigma_{11}\) | \(\sigma_{17}\) | \(\sigma_{16}\) | \(\sigma_5\) | \(\sigma_7\) | \(\sigma_{19}\) | \(\sigma_4\) | \(\sigma_{10}\) | \(\sigma_{22}\) | \(\sigma_{24}\) | \(\sigma_{14}\) | \(\sigma_{13}\) | \(\sigma_{23}\) | \(\sigma_1\) | \(\sigma_9\) | \(\sigma_{21}\) | \(\sigma_{20}\) | \(\sigma_6\) | \(\sigma_{18}\) | \(\sigma_2\) | \(\sigma_3\) | \(\sigma_{15}\) | \(\sigma_{12}\) | \(\sigma_8\) |
| \(\sigma_{12}\) | \(\sigma_{12}\) | \(\sigma_{18}\) | \(\sigma_6\) | \(\sigma_{19}\) | \(\sigma_3\) | \(\sigma_5\) | \(\sigma_{20}\) | \(\sigma_{13}\) | \(\sigma_{23}\) | \(\sigma_1\) | \(\sigma_9\) | \(\sigma_{10}\) | \(\sigma_{22}\) | \(\sigma_{24}\) | \(\sigma_{14}\) | \(\sigma_2\) | \(\sigma_4\) | \(\sigma_{16}\) | \(\sigma_{17}\) | \(\sigma_{21}\) | \(\sigma_7\) | \(\sigma_8\) | \(\sigma_{11}\) | \(\sigma_{15}\) |
| \(\sigma_{13}\) | \(\sigma_{13}\) | \(\sigma_6\) | \(\sigma_{19}\) | \(\sigma_{18}\) | \(\sigma_{16}\) | \(\sigma_7\) | \(\sigma_2\) | \(\sigma_{23}\) | \(\sigma_{12}\) | \(\sigma_{11}\) | \(\sigma_{22}\) | \(\sigma_{24}\) | \(\sigma_{15}\) | \(\sigma_8\) | \(\sigma_1\) | \(\sigma_{17}\) | \(\sigma_5\) | \(\sigma_{21}\) | \(\sigma_{20}\) | \(\sigma_3\) | \(\sigma_4\) | \(\sigma_{10}\) | \(\sigma_{14}\) | \(\sigma_9\) |
| \(\sigma_{14}\) | \(\sigma_{14}\) | \(\sigma_{20}\) | \(\sigma_{21}\) | \(\sigma_7\) | \(\sigma_4\) | \(\sigma_{18}\) | \(\sigma_5\) | \(\sigma_{24}\) | \(\sigma_{15}\) | \(\sigma_8\) | \(\sigma_1\) | \(\sigma_{23}\) | \(\sigma_{12}\) | \(\sigma_{11}\) | \(\sigma_{22}\) | \(\sigma_3\) | \(\sigma_2\) | \(\sigma_{19}\) | \(\sigma_6\) | \(\sigma_{17}\) | \(\sigma_{16}\) | \(\sigma_9\) | \(\sigma_{13}\) | \(\sigma_{10}\) |
| \(\sigma_{15}\) | \(\sigma_{15}\) | \(\sigma_7\) | \(\sigma_{20}\) | \(\sigma_{21}\) | \(\sigma_{17}\) | \(\sigma_2\) | \(\sigma_6\) | \(\sigma_{14}\) | \(\sigma_{24}\) | \(\sigma_{22}\) | \(\sigma_{10}\) | \(\sigma_9\) | \(\sigma_1\) | \(\sigma_{23}\) | \(\sigma_{13}\) | \(\sigma_5\) | \(\sigma_{16}\) | \(\sigma_4\) | \(\sigma_3\) | \(\sigma_{19}\) | \(\sigma_{18}\) | \(\sigma_{11}\) | \(\sigma_8\) | \(\sigma_{12}\) |
| \(\sigma_{16}\) | \(\sigma_{16}\) | \(\sigma_{10}\) | \(\sigma_{11}\) | \(\sigma_{22}\) | \(\sigma_{13}\) | \(\sigma_{24}\) | \(\sigma_8\) | \(\sigma_{17}\) | \(\sigma_5\) | \(\sigma_{19}\) | \(\sigma_{18}\) | \(\sigma_7\) | \(\sigma_{21}\) | \(\sigma_2\) | \(\sigma_3\) | \(\sigma_{23}\) | \(\sigma_{12}\) | \(\sigma_{15}\) | \(\sigma_{14}\) | \(\sigma_1\) | \(\sigma_9\) | \(\sigma_6\) | \(\sigma_{20}\) | \(\sigma_4\) |
| \(\sigma_{17}\) | \(\sigma_{17}\) | \(\sigma_{11}\) | \(\sigma_{22}\) | \(\sigma_{10}\) | \(\sigma_{15}\) | \(\sigma_9\) | \(\sigma_{23}\) | \(\sigma_5\) | \(\sigma_{16}\) | \(\sigma_{20}\) | \(\sigma_{21}\) | \(\sigma_2\) | \(\sigma_4\) | \(\sigma_6\) | \(\sigma_{19}\) | \(\sigma_{14}\) | \(\sigma_{24}\) | \(\sigma_1\) | \(\sigma_8\) | \(\sigma_{13}\) | \(\sigma_{12}\) | \(\sigma_7\) | \(\sigma_3\) | \(\sigma_{18}\) |
| \(\sigma_{18}\) | \(\sigma_{18}\) | \(\sigma_{12}\) | \(\sigma_{23}\) | \(\sigma_{13}\) | \(\sigma_8\) | \(\sigma_{14}\) | \(\sigma_{22}\) | \(\sigma_{19}\) | \(\sigma_6\) | \(\sigma_4\) | \(\sigma_2\) | \(\sigma_{21}\) | \(\sigma_{20}\) | \(\sigma_{16}\) | \(\sigma_5\) | \(\sigma_9\) | \(\sigma_1\) | \(\sigma_{24}\) | \(\sigma_{15}\) | \(\sigma_{10}\) | \(\sigma_{11}\) | \(\sigma_3\) | \(\sigma_7\) | \(\sigma_{17}\) |
| \(\sigma_{19}\) | \(\sigma_{19}\) | \(\sigma_{23}\) | \(\sigma_{13}\) | \(\sigma_{12}\) | \(\sigma_{24}\) | \(\sigma_{11}\) | \(\sigma_9\) | \(\sigma_6\) | \(\sigma_{18}\) | \(\sigma_7\) | \(\sigma_{20}\) | \(\sigma_{16}\) | \(\sigma_{17}\) | \(\sigma_3\) | \(\sigma_4\) | \(\sigma_{15}\) | \(\sigma_{14}\) | \(\sigma_{10}\) | \(\sigma_{22}\) | \(\sigma_8\) | \(\sigma_1\) | \(\sigma_{21}\) | \(\sigma_5\) | \(\sigma_2\) |
| \(\sigma_{20}\) | \(\sigma_{20}\) | \(\sigma_{14}\) | \(\sigma_{15}\) | \(\sigma_{24}\) | \(\sigma_9\) | \(\sigma_{22}\) | \(\sigma_{12}\) | \(\sigma_7\) | \(\sigma_{21}\) | \(\sigma_2\) | \(\sigma_3\) | \(\sigma_{17}\) | \(\sigma_5\) | \(\sigma_{19}\) | \(\sigma_{18}\) | \(\sigma_1\) | \(\sigma_8\) | \(\sigma_{11}\) | \(\sigma_{10}\) | \(\sigma_{23}\) | \(\sigma_{13}\) | \(\sigma_4\) | \(\sigma_{16}\) | \(\sigma_6\) |
| \(\sigma_{21}\) | \(\sigma_{21}\) | \(\sigma_{24}\) | \(\sigma_{14}\) | \(\sigma_{15}\) | \(\sigma_{23}\) | \(\sigma_8\) | \(\sigma_{10}\) | \(\sigma_{20}\) | \(\sigma_7\) | \(\sigma_{18}\) | \(\sigma_6\) | \(\sigma_4\) | \(\sigma_3\) | \(\sigma_{17}\) | \(\sigma_{16}\) | \(\sigma_{12}\) | \(\sigma_{13}\) | \(\sigma_9\) | \(\sigma_1\) | \(\sigma_{11}\) | \(\sigma_{22}\) | \(\sigma_{19}\) | \(\sigma_2\) | \(\sigma_5\) |
| \(\sigma_{22}\) | \(\sigma_{22}\) | \(\sigma_5\) | \(\sigma_{17}\) | \(\sigma_{16}\) | \(\sigma_2\) | \(\sigma_{20}\) | \(\sigma_{18}\) | \(\sigma_{11}\) | \(\sigma_{10}\) | \(\sigma_9\) | \(\sigma_8\) | \(\sigma_{15}\) | \(\sigma_{14}\) | \(\sigma_{13}\) | \(\sigma_{12}\) | \(\sigma_4\) | \(\sigma_3\) | \(\sigma_7\) | \(\sigma_{21}\) | \(\sigma_6\) | \(\sigma_{19}\) | \(\sigma_1\) | \(\sigma_{24}\) | \(\sigma_{23}\) |
| \(\sigma_{23}\) | \(\sigma_{23}\) | \(\sigma_{19}\) | \(\sigma_{18}\) | \(\sigma_6\) | \(\sigma_{21}\) | \(\sigma_4\) | \(\sigma_{17}\) | \(\sigma_{12}\) | \(\sigma_{13}\) | \(\sigma_{14}\) | \(\sigma_{15}\) | \(\sigma_8\) | \(\sigma_9\) | \(\sigma_{10}\) | \(\sigma_{11}\) | \(\sigma_{20}\) | \(\sigma_7\) | \(\sigma_3\) | \(\sigma_2\) | \(\sigma_{16}\) | \(\sigma_5\) | \(\sigma_{24}\) | \(\sigma_1\) | \(\sigma_{22}\) |
| \(\sigma_{24}\) | \(\sigma_{24}\) | \(\sigma_{21}\) | \(\sigma_7\) | \(\sigma_{20}\) | \(\sigma_{19}\) | \(\sigma_{16}\) | \(\sigma_3\) | \(\sigma_{15}\) | \(\sigma_{14}\) | \(\sigma_{13}\) | \(\sigma_{12}\) | \(\sigma_{11}\) | \(\sigma_{10}\) | \(\sigma_9\) | \(\sigma_8\) | \(\sigma_6\) | \(\sigma_{18}\) | \(\sigma_{17}\) | \(\sigma_5\) | \(\sigma_4\) | \(\sigma_2\) | \(\sigma_{23}\) | \(\sigma_{22}\) | \(\sigma_1 \) |
（注）【表2】には \(\sigma_i \circ \sigma_j=\sigma_1 \) となるセルを黄色で色付けしてあります。これによって、 \(S_4\) の各元の逆元が同じ共役類の中にある事がすぐに判る様になっています。 この後の節で（式(5.1)参照）、射影演算子からブロック分解行列 \(Q\) を求める際に \(\sigma_{i}^{-1}\) はどの元になるか？そしてその指標は何か？が重要となります。
その時に、この積表を参考にすると、直ぐに逆元と指標の値が判ると思います。
