Let $k$ be a field, $n$ a positive integer. The group $S_n$ acts on $R_n=k[x_1,\dots,x_n]$ by permuting indices, and $\mathcal{S}_n=R_n^{S_n}=k[s_1,\dots,s_n]$ where the $s_i$'s are the usual elementary symmetric functions of the $x_i$'s. Given a subgroup $H<S_n$, is there a simple condition to determine whether or not $R_n^H$ is monogenic over $\mathcal{S}_n$? E.g. for $H=A_n$, $R_n^{A_n}=\mathcal{S}_n[\delta]$, where $\delta = \prod_{1 \leq i < j \leq n} (x_i - x_j)$ is the discriminant polynomial. More generally, is there a simple determination of the minimal size of a generating set for $R_n^{H}$ over $\mathcal{S}_n$?

  • $\begingroup$ A partial answer. You can compute the Hilbert series of $R_n^H$ using Molien's theorem and divide it by the Hilbert series of $R_n^{S_n}$. The result should be a polynomial describing the degrees of extra elements of $R_n^H$, and examining its smallest terms puts a lower bound on how many generators are needed, e.g. if the smallest terms are $2x^3 + x^4 + ... $ then there are two generators of degree $3$ and one of degree $4$. But this is really only useful for fixed $H$ of small order; I assume you are more interested in $H$ of small index. $\endgroup$ – Qiaochu Yuan Nov 4 '11 at 18:56

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