대칭군의 지표(character)에 대한 프로베니우스 공식

\(C_\mathbf{i}}=(1^{i_1},2^{i_2},\cdots,m^{i_m})\) conjugacy class in \(S_{m}\) where \(i_1+2i_2+\cdots mi_m=m\)
\(\left(\sum_{l=1}^{m} x_l\right)^{i_1}\left(\sum_{l=1}^{m} x_l^2\right)^{i_2}\cdots \left(\sum_{l=1}^{m} x_l^m\right)^{i_m}=\sum_{\lambda}\chi_{\lambda}(C_{\mathbf{i}})S_{\lambda}\)
여기서 \(\chi_{\lambda}\) 는 character, \(S_{\lambda}\) 는 슈르 다항식(Schur polynomial)

\(S_{(3)}=x_1 x_2 x_3+\left(x_1+x_2+x_3\right){}^3-2 \left(x_1+x_2+x_3\right) \left(x_1 x_2+x_1 x_3+x_2 x_3\right)\)

\(S_{(2,1)}=\left(x_1+x_2\right) \left(x_1+x_3\right) \left(x_2+x_3\right)\)

\(S_{(1,1,1)}=x_1 x_2 x_3\)

\(S_{(3)}+2S_{(2,1)}+S_{(1,1,1)}=\left(x_1+x_2+x_3\right){}^3\)

\(S_{(3)}+0\cdot S_{(2,1)}-S_{(1,1,1)}=\left(x_1+x_2+x_3\right) \left(x_1^2+x_2^2+x_3^2\right)\)

\(S_{(3)}-1S_{(2,1)}+S_{(1,1,1)}=x_1^3+x_2^3+x_3^3\)

\(\prod_{j}P_{j}(x)^{i_j}=\sum_{\lambda}\chi_{\lambda}(C_{\mathbf{i}})S_{\lambda}\)

목차