2
$\begingroup$

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 $\mathcal{S}_n$?

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.