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

For Formula: see text an infinite field of characteristic other than two, consider the action of the special orthogonal group Formula: see text on a polynomial ring via copies of the regular representation. When Formula: see text has characteristic zero, Boutot’s theorem implies that the invariant ring has rational singularities; when Formula: see text has positive characteristic, the invariant ring is Formula: see text-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for Formula: see text 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 Formula: see text-invariant and information on the Hilbert series. Indeed, we use this to show that the Formula: see text-vector of the invariant ring for Formula: see text need not be unimodal.