When are the natural embeddings of classical invariant rings pure?

TL;DR

This paper investigates when the natural embedding of classical invariant rings into polynomial rings remains pure in positive characteristic, revealing that purity occurs only when the invariant ring is regular or the group is linearly reductive.

Contribution

It characterizes the purity of invariant ring embeddings in positive characteristic, extending classical results from characteristic zero to new cases.

Findings

Purity of the invariant ring embedding implies the invariant ring is regular.

If the embedding is pure, then the group must be linearly reductive or the invariant ring regular.

The paper provides a complete characterization of purity conditions in positive characteristic.

Abstract

Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Pl\"ucker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of $S$, equivalently, $S^G$ is a direct summand of $S$ as an $S^G$-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular, or the group $G$ is linearly reductive.