Permutation on a Polynomial is a Group Action

Theorem

Let $n \in \Z: n > 0$.

Let $f \left({x_1, x_2, \ldots, x_n}\right)$ be a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$.

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

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

Let $\pi * f$ be the Definition:Permutation on a Polynomial$f$ by $\pi$.

Then:

$(1): \quad e * f = f$

$(2): \quad \pi \rho * f = \pi * \left({\rho * f}\right)$

$(3): \quad \forall \lambda \in \R: \pi * \left({\lambda f}\right) = \lambda \left({\pi * f}\right)$

Thus this is an example of a group action where $S_n$ acts on the set of all polynomials in $n$ variables.

The stabilizer of a polynomial is the set of permutations which fix the given polynomial.

Sources

• John F. Humphreys: A Course in Group Theory (1996): $\S 9$: Proposition $9.13$

• John F. Humphreys: A Course in Group Theory (1996): $\S 10$: Example $10.3, \ 10.10$