Homological properties of invariant rings of permutation groups
Aryaman Maithani
TL;DR
This work analyzes the homological structure of invariant rings $S^G$ under permutation group actions on $S=k[x_1,\dots,x_n]$, revealing characteristic-independent phenomena away from char $2$ and explicit behavior in char $2$. It connects local cohomology, duality, and twisted representations to compare $H^n_\mathfrak m(S)^G$ with $H^n_\mathfrak n(S^G)$, and provides concrete criteria for the $a$-invariant and quasi-Gorenstein property, computable via Molien-type formulas in characteristic zero. In odd characteristic, $H^n_\mathfrak m(S)^G\cong (S^G_{\operatorname{sgn}})^*$ and $\omega_{S^G}$ is described in terms of a sign-twist and a product over transpositions, yielding a characteristic-free criterion for quasi-Gorensteinness and a precise $a$-invariant when $G$ contains transpositions. In characteristic two, $S^G$ is always quasi-Gorenstein with an explicit $a$-invariant and a shift relation between the top local cohomology of $S$ and $S^G$, reflecting the behavior of transvections. The paper also proves the Shank–Wehlau conjecture for permutation subgroups by characterizing when $S^G\hookrightarrow S$ splits, showing it holds exactly when the splitting condition avoids the modular obstruction, and establishes when these invariant rings are direct summands, using the structure of the subgroup generated by transpositions. Overall, the results provide characteristic-uniform methods to compute and compare key invariants of $S^G$, including the Molien formulas for semi-invariants and explicit canonical module descriptions, with implications for splitting properties and Gorenstein-type behavior.
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 and 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. Lastly, we determine when the inclusion $S^{G} \hookrightarrow S$ splits, thereby proving the Shank--Wehlau conjecture for permutation subgroups.