Invariant rings of the special orthogonal group have nonunimodal $h$-vectors

TL;DR

This paper investigates the invariant rings of the special orthogonal group acting on polynomial rings, revealing that their $h$-vectors can be nonunimodal, and provides new proofs and insights into their algebraic properties.

Contribution

It offers a new proof that the invariant ring is $F$-regular by viewing it as a cyclic cover, and demonstrates that the $h$-vector of these rings can be nonunimodal.

Findings

Invariant rings are $F$-regular in positive characteristic.

The $h$-vector of the invariant ring can be nonunimodal.

The cyclic cover perspective yields new algebraic insights.

Abstract

For $K$ an infinite field of characteristic other than two, consider the action of the special orthogonal group $\operatorname{SO}_t(K)$ on a polynomial ring via copies of the regular representation. When $K$ has characteristic zero, Boutot's theorem implies that the invariant ring has rational singularities; when $K$ has positive characteristic, the invariant ring is $F$-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for $\operatorname{SO}_t(K)$ as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example it readily yields the $a$-invariant and information on the Hilbert series. Indeed, we use this to show that the $h$-vector of the invariant ring for $\operatorname{SO}_t(K)$ need not be unimodal.