# Definition:Permutation on Polynomial

## Definition

Let $\map f {x_1, x_2, \ldots, x_n}$ denote a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\pi, \rho \in S_n$.

Then $\pi * f$ is the polynomial obtained by applying the permutation $\pi$ to the subscripts on the variables of $f$.

This is called the **permutation on the polynomial $f$ by $\pi$**, or the $f$-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

- Results about
**permutations on polynomials**can be found here.

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $9$: Permutations: Proposition $9.11$