Polynomial with given group of symmetries

Question

Let $f$ be a polynomial in $n$ variables and $G$ - its group of symmetries (group of permutations of variables wich left $f$ in place). I'm trying to such $f$ for given group $G$. I have troubles when $G\subset S_5$ is a cyclic group of fifth order. I have a fifth degree polynomial (for example, $x_1^2x_3x_4x_5+x_2^2x_1x_4x_5+x_3^2x_1x_2x_5+x_4^2x_1x_2x_3+x_5^2x_2x_3x_4$). Is it possible to find such $f$ with lower degree?

Answer 1

Certainly $f$ cannot be linear: if $f = \displaystyle \sum_{i=1}^5 a_i x_i$ is invariant under cyclic permutation then the $a_i = a$ must all be equal to each other, and then $f$ is invariant under all permutations.

$f$ can be quadratic: let $f = \displaystyle \sum_{i=1}^5 (x_i x_{i+1} + 2 x_i x_{i+2})$ where the indices are considered $\bmod 5$.

The general question of whether such polynomials always exist can be addressed using representation theory. The problem reduces to showing that if $G$ is a finite group acting on a vector space $V$ and $H$ is a proper subgroup of $G$, then there exists some $d$ such that $\dim S^d(V)^G$ is strictly less than $\dim S^d(V)^H$, and this is a corollary of Molien's theorem. One then applies this theorem to some fixed group $H \subseteq S_n$, where $V$ is the permutation representation of $S_n$, and $G$ runs over all minimal subgroups of $S_n$ properly containing $H$.