Syzygies of the transfer ideal of the symmetric group

Abstract

We consider the modular action of the symmetric group $S_n$ on $R=k[x_1,\dots ,x_n]$ when $\mathrm{char}(k)=p\le n$. We show that the image of the transfer map $R\to R^{S_n}$ is an elimination ideal $J\cap R^{S_n}$, where $J\subset R^{S_n}[t]$ is generated by $p$ polynomials with generic coefficients. The structure of this elimination ideal depends only on the quotient $q$ when writing $n=qp+r$ with unique remainder $0\le r<p$, implying that the image of the transfer also enjoys this stability. We conjecture a determinantal presentation of the elimination ideal and prove it in the case that $q=2$. Furthermore, we exhibit a GL-equivariant, linear minimal free resolution of a certain initial ideal, allowing us to extract the graded Betti numbers of the elimination ideal.

19 pages, comments welcome