【第2章】対称群 \(S_4\)
【2-1】\(S_4\)の元と番号付け及び群代数\(\mathbb{C}[S_4]\)の元との対応
【表1】は、本サイトで扱う対称群 \(S_4\) の群の番号とその元の置換の内容を対応させたものです。番号付けは巡回置換の長さが \([1,2,3,4]\) と順番に大きくなっております。
更に【第1章】でも説明した群代数 \(\mathbb{C}[S_4]\) の基底との対応も載せておきます。
| \(S_4\) の元 | \(\sigma_{1}\) | \(\sigma_{2}\) | \(\sigma_{3}\) | \(\sigma_{4}\) | \(\sigma_{5}\) | \(\sigma_{6}\) |
|---|---|---|---|---|---|---|
| 2行表現 | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 2 & 3 & 4\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 2 & 4 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 3 & 2 & 4\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 4 & 3 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 1 & 3 & 4\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 2 & 1 & 4\end{pmatrix} \) |
| 巡回表現 | \( e\) | \( (3,4) \) | \( (2,3) \) | \( (2,4) \) | \( (1,2) \) | \( (1,3) \) |
| 群代数 \(\mathbb{C}[S_4]\) の元 | \( g_{1}\) | \( g_{2} \) | \( g_{3} \) | \( g_{4} \) | \(g_{5} \) | \( g_{6} \) |
| \(S_4\) の元 | \(\sigma_{7}\) | \(\sigma_{8}\) | \(\sigma_{9}\) | \(\sigma_{10}\) | \(\sigma_{11}\) | \(\sigma_{12}\) |
|---|---|---|---|---|---|---|
| 2行表現 | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 2 & 3 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 3 & 4 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\1 & 4 & 2 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 3 & 1 & 4\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 4 & 3 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 1 & 2 & 4\end{pmatrix} \) |
| 巡回表現 | \( (1,4) \) | \( (2,3,4) \) | \( (2,4,3) \) | \( (1,2,3) \) | \( (1,2,4) \) | \( (1,3,2) \) |
| 群代数 \(\mathbb{C}[S_4]\) の元 | \( g_{7}\) | \( g_{8} \) | \( g_{9} \) | \( g_{10} \) | \(g_{11} \) | \( g_{12} \) |
| \(S_4\) の元 | \(\sigma_{13}\) | \(\sigma_{14}\) | \(\sigma_{15}\) | \(\sigma_{16}\) | \(\sigma_{17}\) | \(\sigma_{18}\) |
|---|---|---|---|---|---|---|
| 2行表現 | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 2 & 4 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 1 & 3 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 2 & 1 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 3 & 4 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 4 & 1 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 1 & 4 & 2\end{pmatrix} \) |
| 巡回表現 | \( (1,3,4) \) | \( (1,4,2) \) | \( (1,4,3) \) | \( (1,2,3,4) \) | \( (1,2,4,3) \) | \( (1,3,4,2) \) |
| 群代数 \(\mathbb{C}[S_4]\) の元 | \( g_{13}\) | \( g_{14} \) | \( g_{15} \) | \( g_{16} \) | \(g_{17} \) | \( g_{18} \) |
| \(S_4\) の元 | \(\sigma_{19}\) | \(\sigma_{20}\) | \(\sigma_{21}\) | \(\sigma_{22}\) | \(\sigma_{23}\) | \(\sigma_{24}\) |
|---|---|---|---|---|---|---|
| 2行表現 | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 4 & 2 & 1\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 1 & 2 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 3 & 1 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\2 & 1 & 4 & 3\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\3 & 4 & 1 & 2\end{pmatrix} \) | \( \begin{pmatrix}1 & 2 & 3 & 4\\4 & 3 & 2 & 1\end{pmatrix} \) |
| 巡回表現 | \( (1,3,2,4) \) | \( (1,4,3,2) \) | \( (1,4,2,3) \) | \( (1,2)(3,4) \) | \( (1,3)(2,4) \) | \( (1,4)(2,3) \) |
| 群代数 \(\mathbb{C}[S_4]\) の元 | \( g_{19}\) | \( g_{20} \) | \( g_{21} \) | \( g_{22} \) | \(g_{23} \) | \( g_{24} \) |
【2-2】\(\mathbb{C}[S_4]\) の導入及び基底の積表
群代数 \(\mathbb{C}[S_4]\) は、体 \(\mathbb{C}\) 上の \(S_4\) の元を基底とするベクトル空間で、任意の元 \(x\) は(2.1)の様に定式化されます。演算規則は(2.2)～(2.4)の様になっております。基底同士の積は \(S_4\) の元の積と全く同一で、【表2】にその積表を載せておきます。
\begin{align} &x:=a_1 \cdot g_1+ a_2 \cdot g_2 +....+ a_{24} \cdot g_{24}=\displaystyle \sum_{i=1}^{24} a_i g_i \quad \bigl(a_i \in \mathbb{C},\quad x, \ g_i \in \mathbb{C}[S_4] \bigr) \\ \notag \\ &\qquad (1) \quad \displaystyle \sum_{i=1}^{24} a_i g_i+\displaystyle \sum_{i=1}^{24} b_i g_i=\displaystyle \sum_{i=1}^{24}( a_i+b_i) g_i \quad \bigl(b_i \in \mathbb{C} \bigr)\\ \notag \\ &\qquad (2) \quad c \cdot \bigl( \displaystyle \sum_{i=1}^{24} a_i g_i \bigr)= \displaystyle \sum_{i=1}^{24}(c \cdot a_i) g_i \quad \bigl(c \in \mathbb{C} \bigr)\\ \notag \\ &\qquad (3) \quad \biggl( \displaystyle \sum_{i=1}^{24} a_i g_i \biggr) \cdot \biggl(\displaystyle \sum_{j=1}^{24} b_j g_j \biggr)=\displaystyle \sum_{i,j=1}^{24}( a_i \cdot b_j) g_i \cdot g_j \\ \end{align}
| \(i \backslash j\) | \(g_1\) | \(g_2\) | \(g_3\) | \(g_4\) | \(g_5\) | \(g_6\) | \(g_7\) | \(g_8\) | \(g_9\) | \(g_{10}\) | \(g_{11}\) | \(g_{12}\) | \(g_{13}\) | \(g_{14}\) | \(g_{15}\) | \(g_{16}\) | \(g_{17}\) | \(g_{18}\) | \(g_{19}\) | \(g_{20}\) | \(g_{21}\) | \(g_{22}\) | \(g_{23}\) | \(g_{24}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(g_{1}\) | \(g_1\) | \(g_2\) | \(g_3\) | \(g_4\) | \(g_5\) | \(g_6\) | \(g_7\) | \(g_8\) | \(g_9\) | \(g_{10}\) | \(g_{11}\) | \(g_{12}\) | \(g_{13}\) | \(g_{14}\) | \(g_{15}\) | \(g_{16}\) | \(g_{17}\) | \(g_{18}\) | \(g_{19}\) | \(g_{20}\) | \(g_{21}\) | \(g_{22}\) | \(g_{23}\) | \(g_{24}\) |
| \(g_{2}\) | \(g_2\) | \(g_1\) | \(g_9\) | \(g_8\) | \(g_{22}\) | \(g_{15}\) | \(g_{13}\) | \(g_4\) | \(g_3\) | \(g_{17}\) | \(g_{16}\) | \(g_{20}\) | \(g_7\) | \(g_{18}\) | \(g_6\) | \(g_{11}\) | \(g_{10}\) | \(g_{14}\) | \(g_{24}\) | \(g_{12}\) | \(g_{23}\) | \(g_5\) | \(g_{21}\) | \(g_{19}\) |
| \(g_{3}\) | \(g_3\) | \(g_8\) | \(g_1\) | \(g_9\) | \(g_{12}\) | \(g_{10}\) | \(g_{24}\) | \(g_2\) | \(g_4\) | \(g_6\) | \(g_{19}\) | \(g_5\) | \(g_{16}\) | \(g_{20}\) | \(g_{21}\) | \(g_{13}\) | \(g_{23}\) | \(g_{22}\) | \(g_{11}\) | \(g_{14}\) | \(g_{15}\) | \(g_{18}\) | \(g_{17}\) | \(g_7\) |
| \(g_{4}\) | \(g_4\) | \(g_9\) | \(g_8\) | \(g_1\) | \(g_{14}\) | \(g_{23}\) | \(g_{11}\) | \(g_3\) | \(g_2\) | \(g_{21}\) | \(g_7\) | \(g_{18}\) | \(g_{19}\) | \(g_5\) | \(g_{17}\) | \(g_{24}\) | \(g_{15}\) | \(g_{12}\) | \(g_{13}\) | \(g_{22}\) | \(g_{10}\) | \(g_{20}\) | \(g_6\) | \(g_{16}\) |
| \(g_{5}\) | \(g_5\) | \(g_{22}\) | \(g_{10}\) | \(g_{11}\) | \(g_1\) | \(g_{12}\) | \(g_{14}\) | \(g_{16}\) | \(g_{17}\) | \(g_3\) | \(g_4\) | \(g_6\) | \(g_{18}\) | \(g_7\) | \(g_{20}\) | \(g_8\) | \(g_9\) | \(g_{13}\) | \(g_{23}\) | \(g_{15}\) | \(g_{24}\) | \(g_2\) | \(g_{19}\) | \(g_{21}\) |
| \(g_{6}\) | \(g_6\) | \(g_{13}\) | \(g_{12}\) | \(g_{23}\) | \(g_{10}\) | \(g_1\) | \(g_{15}\) | \(g_{18}\) | \(g_{19}\) | \(g_5\) | \(g_{17}\) | \(g_3\) | \(g_2\) | \(g_{21}\) | \(g_7\) | \(g_{22}\) | \(g_{11}\) | \(g_8\) | \(g_9\) | \(g_{24}\) | \(g_{14}\) | \(g_{16}\) | \(g_4\) | \(g_{20}\) |
| \(g_{7}\) | \(g_7\) | \(g_{15}\) | \(g_{24}\) | \(g_{14}\) | \(g_{11}\) | \(g_{13}\) | \(g_1\) | \(g_{21}\) | \(g_{20}\) | \(g_{16}\) | \(g_5\) | \(g_{19}\) | \(g_6\) | \(g_4\) | \(g_2\) | \(g_{10}\) | \(g_{22}\) | \(g_{23}\) | \(g_{12}\) | \(g_9\) | \(g_8\) | \(g_{17}\) | \(g_{18}\) | \(g_3\) |
| \(g_{8}\) | \(g_8\) | \(g_3\) | \(g_4\) | \(g_2\) | \(g_{18}\) | \(g_{21}\) | \(g_{16}\) | \(g_9\) | \(g_1\) | \(g_{23}\) | \(g_{13}\) | \(g_{14}\) | \(g_{24}\) | \(g_{22}\) | \(g_{10}\) | \(g_{19}\) | \(g_6\) | \(g_{20}\) | \(g_7\) | \(g_5\) | \(g_{17}\) | \(g_{12}\) | \(g_{15}\) | \(g_{11}\) |
| \(g_{9}\) | \(g_9\) | \(g_4\) | \(g_2\) | \(g_3\) | \(g_{20}\) | \(g_{17}\) | \(g_{19}\) | \(g_1\) | \(g_8\) | \(g_{15}\) | \(g_{24}\) | \(g_{22}\) | \(g_{11}\) | \(g_{12}\) | \(g_{23}\) | \(g_7\) | \(g_{21}\) | \(g_5\) | \(g_{16}\) | \(g_{18}\) | \(g_6\) | \(g_{14}\) | \(g_{10}\) | \(g_{13}\) |
| \(g_{10}\) | \(g_{10}\) | \(g_{16}\) | \(g_5\) | \(g_{17}\) | \(g_6\) | \(g_3\) | \(g_{21}\) | \(g_{22}\) | \(g_{11}\) | \(g_{12}\) | \(g_{23}\) | \(g_1\) | \(g_8\) | \(g_{15}\) | \(g_{24}\) | \(g_{18}\) | \(g_{19}\) | \(g_2\) | \(g_4\) | \(g_7\) | \(g_{20}\) | \(g_{13}\) | \(g_9\) | \(g_{14}\) |
| \(g_{11}\) | \(g_{11}\) | \(g_{17}\) | \(g_{16}\) | \(g_5\) | \(g_7\) | \(g_{19}\) | \(g_4\) | \(g_{10}\) | \(g_{22}\) | \(g_{24}\) | \(g_{14}\) | \(g_{13}\) | \(g_{23}\) | \(g_1\) | \(g_9\) | \(g_{21}\) | \(g_{20}\) | \(g_6\) | \(g_{18}\) | \(g_2\) | \(g_3\) | \(g_{15}\) | \(g_{12}\) | \(g_8\) |
| \(g_{12}\) | \(g_{12}\) | \(g_{18}\) | \(g_6\) | \(g_{19}\) | \(g_3\) | \(g_5\) | \(g_{20}\) | \(g_{13}\) | \(g_{23}\) | \(g_1\) | \(g_9\) | \(g_{10}\) | \(g_{22}\) | \(g_{24}\) | \(g_{14}\) | \(g_2\) | \(g_4\) | \(g_{16}\) | \(g_{17}\) | \(g_{21}\) | \(g_7\) | \(g_8\) | \(g_{11}\) | \(g_{15}\) |
| \(g_{13}\) | \(g_{13}\) | \(g_6\) | \(g_{19}\) | \(g_{18}\) | \(g_{16}\) | \(g_7\) | \(g_2\) | \(g_{23}\) | \(g_{12}\) | \(g_{11}\) | \(g_{22}\) | \(g_{24}\) | \(g_{15}\) | \(g_8\) | \(g_1\) | \(g_{17}\) | \(g_5\) | \(g_{21}\) | \(g_{20}\) | \(g_3\) | \(g_4\) | \(g_{10}\) | \(g_{14}\) | \(g_9\) |
| \(g_{14}\) | \(g_{14}\) | \(g_{20}\) | \(g_{21}\) | \(g_7\) | \(g_4\) | \(g_{18}\) | \(g_5\) | \(g_{24}\) | \(g_{15}\) | \(g_8\) | \(g_1\) | \(g_{23}\) | \(g_{12}\) | \(g_{11}\) | \(g_{22}\) | \(g_3\) | \(g_2\) | \(g_{19}\) | \(g_6\) | \(g_{17}\) | \(g_{16}\) | \(g_9\) | \(g_{13}\) | \(g_{10}\) |
| \(g_{15}\) | \(g_{15}\) | \(g_7\) | \(g_{20}\) | \(g_{21}\) | \(g_{17}\) | \(g_2\) | \(g_6\) | \(g_{14}\) | \(g_{24}\) | \(g_{22}\) | \(g_{10}\) | \(g_9\) | \(g_1\) | \(g_{23}\) | \(g_{13}\) | \(g_5\) | \(g_{16}\) | \(g_4\) | \(g_3\) | \(g_{19}\) | \(g_{18}\) | \(g_{11}\) | \(g_8\) | \(g_{12}\) |
| \(g_{16}\) | \(g_{16}\) | \(g_{10}\) | \(g_{11}\) | \(g_{22}\) | \(g_{13}\) | \(g_{24}\) | \(g_8\) | \(g_{17}\) | \(g_5\) | \(g_{19}\) | \(g_{18}\) | \(g_7\) | \(g_{21}\) | \(g_2\) | \(g_3\) | \(g_{23}\) | \(g_{12}\) | \(g_{15}\) | \(g_{14}\) | \(g_1\) | \(g_9\) | \(g_6\) | \(g_{20}\) | \(g_4\) |
| \(g_{17}\) | \(g_{17}\) | \(g_{11}\) | \(g_{22}\) | \(g_{10}\) | \(g_{15}\) | \(g_9\) | \(g_{23}\) | \(g_5\) | \(g_{16}\) | \(g_{20}\) | \(g_{21}\) | \(g_2\) | \(g_4\) | \(g_6\) | \(g_{19}\) | \(g_{14}\) | \(g_{24}\) | \(g_1\) | \(g_8\) | \(g_{13}\) | \(g_{12}\) | \(g_7\) | \(g_3\) | \(g_{18}\) |
| \(g_{18}\) | \(g_{18}\) | \(g_{12}\) | \(g_{23}\) | \(g_{13}\) | \(g_8\) | \(g_{14}\) | \(g_{22}\) | \(g_{19}\) | \(g_6\) | \(g_4\) | \(g_2\) | \(g_{21}\) | \(g_{20}\) | \(g_{16}\) | \(g_5\) | \(g_9\) | \(g_1\) | \(g_{24}\) | \(g_{15}\) | \(g_{10}\) | \(g_{11}\) | \(g_3\) | \(g_7\) | \(g_{17}\) |
| \(g_{19}\) | \(g_{19}\) | \(g_{23}\) | \(g_{13}\) | \(g_{12}\) | \(g_{24}\) | \(g_{11}\) | \(g_9\) | \(g_6\) | \(g_{18}\) | \(g_7\) | \(g_{20}\) | \(g_{16}\) | \(g_{17}\) | \(g_3\) | \(g_4\) | \(g_{15}\) | \(g_{14}\) | \(g_{10}\) | \(g_{22}\) | \(g_8\) | \(g_1\) | \(g_{21}\) | \(g_5\) | \(g_2\) |
| \(g_{20}\) | \(g_{20}\) | \(g_{14}\) | \(g_{15}\) | \(g_{24}\) | \(g_9\) | \(g_{22}\) | \(g_{12}\) | \(g_7\) | \(g_{21}\) | \(g_2\) | \(g_3\) | \(g_{17}\) | \(g_5\) | \(g_{19}\) | \(g_{18}\) | \(g_1\) | \(g_8\) | \(g_{11}\) | \(g_{10}\) | \(g_{23}\) | \(g_{13}\) | \(g_4\) | \(g_{16}\) | \(g_6\) |
| \(g_{21}\) | \(g_{21}\) | \(g_{24}\) | \(g_{14}\) | \(g_{15}\) | \(g_{23}\) | \(g_8\) | \(g_{10}\) | \(g_{20}\) | \(g_7\) | \(g_{18}\) | \(g_6\) | \(g_4\) | \(g_3\) | \(g_{17}\) | \(g_{16}\) | \(g_{12}\) | \(g_{13}\) | \(g_9\) | \(g_1\) | \(g_{11}\) | \(g_{22}\) | \(g_{19}\) | \(g_2\) | \(g_5\) |
| \(g_{22}\) | \(g_{22}\) | \(g_5\) | \(g_{17}\) | \(g_{16}\) | \(g_2\) | \(g_{20}\) | \(g_{18}\) | \(g_{11}\) | \(g_{10}\) | \(g_9\) | \(g_8\) | \(g_{15}\) | \(g_{14}\) | \(g_{13}\) | \(g_{12}\) | \(g_4\) | \(g_3\) | \(g_7\) | \(g_{21}\) | \(g_6\) | \(g_{19}\) | \(g_1\) | \(g_{24}\) | \(g_{23}\) |
| \(g_{23}\) | \(g_{23}\) | \(g_{19}\) | \(g_{18}\) | \(g_6\) | \(g_{21}\) | \(g_4\) | \(g_{17}\) | \(g_{12}\) | \(g_{13}\) | \(g_{14}\) | \(g_{15}\) | \(g_8\) | \(g_9\) | \(g_{10}\) | \(g_{11}\) | \(g_{20}\) | \(g_7\) | \(g_3\) | \(g_2\) | \(g_{16}\) | \(g_5\) | \(g_{24}\) | \(g_1\) | \(g_{22}\) |
| \(g_{24}\) | \(g_{24}\) | \(g_{21}\) | \(g_7\) | \(g_{20}\) | \(g_{19}\) | \(g_{16}\) | \(g_3\) | \(g_{15}\) | \(g_{14}\) | \(g_{13}\) | \(g_{12}\) | \(g_{11}\) | \(g_{10}\) | \(g_9\) | \(g_8\) | \(g_6\) | \(g_{18}\) | \(g_{17}\) | \(g_5\) | \(g_4\) | \(g_2\) | \(g_{23}\) | \(g_{22}\) | \(g_1 \) |
