Homological properties of invariant rings of permutation groups

TL;DR

The paper investigates the homological properties of invariant rings under permutation groups, showing characteristic independence of key invariants in odd characteristic and providing explicit computations and conditions in characteristic two.

Contribution

It establishes characteristic-independent properties of invariant rings, proves the Shank–Wehlau conjecture for permutation groups, and analyzes differential operators on these rings.

Findings

In characteristic not two, invariants are independent of the field characteristic.

In characteristic two, the invariant ring is always quasi-Gorenstein and the $a$-invariant is explicitly computed.

The inclusion of invariants into the polynomial ring splits under certain conditions, confirming the Shank–Wehlau conjecture.

Abstract

Consider the action of a subgroup $G$ of the permutation group on the polynomial ring $S := k[x_{1}, \ldots, x_{n}]$ via permutations. We show that if $k$ does not have characteristic two, then the following are independent of $k$: the $a$-invariant of $S^{G}$, the property of $S^{G}$ being quasi-Gorenstein, and the Hilbert functions of $H_{\mathfrak{m}}^{n}(S)^{G}$ as well as $H_{\mathfrak{n}}^{n}(S^{G})$; moreover, these Hilbert functions coincide. In particular, being independent of characteristic, they may be computed using characteristic zero techniques, such as Molien's formula. In characteristic two, we show that the ring of invariants is always quasi-Gorenstein, compute the $a$-invariant explicitly, and show that the Hilbert functions of $H_{\mathfrak{m}}^{n}(S)^{G}$ and $H_{\mathfrak{n}}^{n}(S^{G})$ agree up to a shift, given by the number of transpositions. We determine when the inclusion $S^{G} \hookrightarrow S$ splits, thereby proving the Shank--Wehlau conjecture for permutation subgroups. Lastly, we determine the ring of $k$-linear differential operators on $S^{G}$, and show that each differential operator lifts to one over $\mathbb{Z}$.