The symmetric group (=permutation group) $S_n$ acts on the set $X_n$ of polynomials in $n$ variables $x_1, x_2, \cdots, x_n$ [with coefficients from $\mathbb{Z}/ \mathbb{Q}/$ or any ring of characteristic $\neq 2$] by $$\sigma.f(x_1,x_2,\cdots, x_n)=f(x_{\sigma(1)}, x_{\sigma(2)}, \cdots, x_{\sigma(n)}).$$

The alternating group $A_n$ is the stabilizer of the polynomial $$P(x_1,\cdots ,x_n)=\prod_{1\leq i<j\leq n} (x_i-x_j).$$

Question: Can we obtain Sylow subgroup(s) of $S_n$ as stabilizer of some polynomial $g(x_1,x_2,\cdots, x_n)$?


You can obtain every subgroup $U\le S_n$ as stabilizer of a polynomial (but the polynomial is not really that interesting):

Let $$p=\sum_{g\in U} (x_1x_2^2x_3^3\cdots x_n^n)^g.$$ Then $p$ is clearly invariant under $U$, but any element outside $U$ will map $x_1x_2^2x_3^3\cdots x_n^n$ to a term that is not a summand of $p$, so $U$ is the stabilizer of $p$.

I suspect (but don't have an interesting example at hand) that this cannot be improved for arbitrary subgroups of $S_n$, though in many concrete cases there are nicer polynomials.

Finding good (i.e. low degree, few terms) polynomials for particular groups is a nontrivial task.


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