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【第4章】Frobenius群 \(F_{20}\)

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【4-10】 \(F_{20}\) の既約表現のまとめ

Frobenius群 \(F_{20}\) の正則表現を、前節で求めた \(T\) によって既約表現に 分解したときの、最終的な分割された表現に(10.1)の様に \( \rho_{i,j} \) と名前を付けます。

\begin{align} & \widetilde{L_i}=T^{-1} \times L_i \times T= \begin{pmatrix} \boxed{\rho_1}&0&0&0&0&0&0&0\\ 0 & \boxed{\rho_2 }& 0 & 0 & 0 &0&0&0 \\ 0 &0& \boxed{\rho_3 }& 0 & 0 & 0 &0&0 \\ 0 &0&0& \boxed{\rho_4 }& 0 & 0 & 0 &0 \\ 0&0&0&0& \boxed{ \begin{matrix} & & \\ & \rho_{5,1} & \\ & 4 \times 4 & \end{matrix}} & 0 & 0 &0\\ 0&0&0&0&0& \boxed{ \begin{matrix} & & \\ & \rho_{5,2} & \\ & 4 \times 4 & \end{matrix}} & 0 & 0\\ 0&0&0&0&0&0&\boxed{ \begin{matrix} & & \\ & \rho_{5,3} & \\ & 4 \times 4 & \end{matrix}} & 0\\ 0&0&0&0&0&0&0& \boxed{ \begin{matrix} & & \\ & \rho_{5,4} & \\ & 4 \times 4 & \end{matrix}}\\ \end{pmatrix} \\ \end{align}



冗長になるかもしれませんが、既約表現に分解された(20x20)の大きな行列 \(\widetilde{L_i}\) 全てを記述することは 出来ません。しかし、既約表現から8種類の小ブロックを抽出した行列ならば、全てを表示できます。そこで、(10.1)の表示 \(\rho_{i,j}\) と 対応付けて整理したのが下記の表となります。何かの参考になれば幸いです。

【表4】\(F_{20}\) の元の番号付け
\(F_{20}\) の元\(\sigma_{1}\)\(\sigma_{2}\)\(\sigma_{3}\)\(\sigma_{4}\)\(\sigma_{5}\)
巡回置換\(e\)\((12345)\)\((13524)\)\((14253)\)\((15432)\)
\(\rho_{1}\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(\rho_{2}\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(\rho_{3}\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(\rho_{4}\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(\rho_{5,1}\)\(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\end{bmatrix}\)
\(\rho_{5,2}\)\(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\end{bmatrix}\)
\(\rho_{5,3}\)\(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\end{bmatrix}\)
\(\rho_{5,4}\)\(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\end{bmatrix}\)

\(F_{20}\) の元\(\sigma_{6}\)\(\sigma_{7}\)\(\sigma_{8}\)\(\sigma_{9}\)\(\sigma_{10}\)
巡回置換\((2354)\)\((1243)\)\((1325)\)\((1452)\)\((1534)\)
\(\rho_{1}\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(\rho_{2}\)\(i\)\(i\)\(i\)\(i\)\(i\)
\(\rho_{3}\)\(-1\)\(-1\)\(-1\)\(-1\)\(-1\)
\(\rho_{4}\)\(-i\)\(-i\)\(-i\)\(-i\)\(-i\)
\(\rho_{5,1}\)\(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\end{bmatrix}\)
\(\rho_{5,2}\)\(\begin{bmatrix}0 & 0 & 0 & 1\\0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 1 & 0 & 0\end{bmatrix}\)
\(\rho_{5,3}\)\(\begin{bmatrix}0 & 0 & 1 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\\1 & 0 & 0 & 0\end{bmatrix}\)
\(\rho_{5,4}\)\(\begin{bmatrix}0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 0 & 1 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 1 & 0 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\)

\(F_{20}\) の元\(\sigma_{11}\)\(\sigma_{12}\)\(\sigma_{13}\)\(\sigma_{14}\)\(\sigma_{15}\)
巡回置換\((25)(34)\)\((14)(23)\)\((12)(35)\)\((15)(24)\)\((13)(45)\)
\(\rho_{1}\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(\rho_{2}\)\(-1\)\(-1\)\(-1\)\(-1\)\(-1\)
\(\rho_{3}\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(\rho_{4}\)\(-1\)\(-1\)\(-1\)\(-1\)\(-1\)
\(\rho_{5,1}\)\(\begin{bmatrix}0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\)
\(\rho_{5,2}\)\(\begin{bmatrix}0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}\)
\(\rho_{5,3}\)\(\begin{bmatrix}0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\end{bmatrix}\)
\(\rho_{5,4}\)\(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{bmatrix}\)

\(F_{20}\) の元\(\sigma_{16}\)\(\sigma_{17}\)\(\sigma_{18}\)\(\sigma_{19}\)\(\sigma_{20}\)
巡回置換\((2453)\)\((1342)\)\((1523)\)\((1254)\)\((1435)\)
\(\rho_{1}\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(\rho_{2}\)\(-i\)\(-i\)\(-i\)\(-i\)\(-i\)
\(\rho_{3}\)\(-1\)\(-1\)\(-1\)\(-1\)\(-1\)
\(\rho_{4}\)\(i\)\(i\)\(i\)\(i\)\(i\)
\(\rho_{5,1}\)\(\begin{bmatrix}0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\end{bmatrix}\)
\(\rho_{5,2}\)\(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}\)
\(\rho_{5,3}\)\(\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\)
\(\rho_{5,4}\)\(\begin{bmatrix}0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\end{bmatrix}\) \(\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}\) \(\begin{bmatrix}-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\end{bmatrix}\) \(\begin{bmatrix}0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\end{bmatrix}\) \(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\-1 & -1 & -1 & -1\\0 & 1 & 0 & 0\end{bmatrix}\)

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home \(\quad \)
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