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.

Examples

Polynomial on 3 Variables

Consider the polynomial on $3$ variables:

$\map f {x_1, x_2, x_3} = {x_1}^2 + 2 x_1 x_2 = 4 x_1 x_2 {x_3}^2$

Let $\rho := \begin{pmatrix} 1 & 2 & 3 \end{pmatrix}$ be a permutation on the Symmetric Group on 3 Letters $S_3$.

Then:

$\rho \circ f = {x_2}^2 + 2 x_2 x_3 = 4 x_2 x_3 {x_1}^2$

Also see

• Permutation on Polynomial is Group Action

• Results about permutations on polynomials can be found here.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 81$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Proposition $9.11$