Definition:Permutation on Polynomial

Definition
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 $\pi$.

Also known as
This is also called the permutation of the polynomial.

Also see

 * Permutation on Polynomial is Group Action