シューア多項式
x n d n ] = ∑ σ ∈ S n ε ( σ ) x σ ( 1 ) d 1 ⋯ x σ ( n ) d n {\displaystyle a_{(d_{1},d_{2},\dots ,d_{n})}(x_{1},x_{2},\dots ,x_{n})=\det \left[{\begin{matrix}x_{1}^{d_{1}}&x_{2}^{d_{1}}&\dots &x_{n}^{d_{1}}\\x_{1}^{d_{2}}&x_{2}^{d_{2}}&\dots &x_{n}^{d_{2}}\\\vdots &\vdots &\ddots &\vdots \\x_{1}^{d_{n}}&x_{2}^{d_{n}}&\dots &x_{n}^{d_{n}}\end{matrix}}\right]=\sum _{\sigma \in S_{n}}\epsilon (\sigma )x_{\sigma (1)}^{d_{1}}\cdots x_{\sigma (n)}^{d_{n}}}
が定まる。交代式であることから、ファンデルモンド行列式
a ( n − 1 , n − 2 , ... , 0 ) ( x 1 , x 2 , ... , x n ) = det [ x 1 n − 1 x 2 n − 1 ... x n n − 1 x 1 n − 2 x 2 n − 2 ... x n n − 2 ⋮ ⋮ ⋱ ⋮ 1 1 ... 1 ] = ∏ 1 ≤ j < k ≤ n ( x j − x k ) . {\displaystyle a_{(n-1,n-2,\dots ,0)}(x_{1},x_{2},\dots ,x_{n})=\det \left[{\begin{matrix}x_{1}^{n-1}&x_{2}^{n-1}&\dots &x_{n}^{n-1}\\x_{1}^{n-2}&x_{2}^{n-2}&\dots &x_{n}^{n-2}\\\vdots &\vdots &\ddots &\vdots \\1&1&\dots &1\end{matrix}}\right]=\prod _{1\leq j<k\leq n}(x_{j}-x_{k}).