\begin{definition}[Definition:Permutation on Polynomial]
Let $\map {\mathscr P_n} {x_1, x_2, \ldots, x_n}$ denote a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $S_n$ be the group action on $\mathscr P_n$ defined as follows.
Let $\pi \in S_n$.
Then $\pi * \mathscr P_n$ is the polynomial obtained by applying the permutation $\pi$ to the subscripts on the variables of $\mathscr P_n$.
This is called the '''permutation on the polynomial $\mathscr P_n$ by $\pi$''', or the '''$\mathscr P_n$-permutation by $\pi$'''.
\end{definition}