Syzygies of the transfer ideal of the symmetric group

TL;DR

This paper studies the transfer ideal of the symmetric group acting on a polynomial ring in characteristic p, revealing its structure, stability, and providing a minimal free resolution in certain cases.

Contribution

It introduces a new understanding of the transfer ideal's structure, including a conjecture on its determinantal presentation and explicit resolutions in specific cases.

Findings

The transfer map image is an elimination ideal generated by p polynomials.

The structure depends only on the quotient n = qp + r, showing stability across parameters.

A GL-equivariant minimal free resolution of an initial ideal is constructed, revealing Betti numbers.

Abstract

We consider the modular action of the symmetric group $S_n$ on $R = k[x_1,\ldots,x_n]$ when $\mathrm{char}(k) = p \leq n$. 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 $0 \leq r < p$, 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.