The Frobenius morphism in invariant theory II

TL;DR

This paper describes the decomposition of the coordinate ring of the Grassmannian as a module over its Frobenius powers, demonstrating finite F-representation type and related properties for a non-linearly reductive group invariant ring.

Contribution

It provides the first non-trivial example of a ring of invariants with finite F-representation type for a non-linearly reductive group, extending previous results to higher Frobenius powers.

Findings

R has finite F-representation type (FFRT).

The ring of differential operators D_k(R) is simple.

G has global finite F-representation type (GFFRT).

Abstract

Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=Gr(2,n)$ defined over an algebraically closed field $k$ of characteristic $p \geq \max\{n-2,3\}$. In this paper we give a description of the decomposition of $R$, considered as graded $R^{p^r}$-module, for $r \geq 2$. This is a companion paper to our earlier paper, where the case $r=1$ was treated, and taken together, our results imply that $R$ has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators $D_k(R)$ is simple, that $\mathbb{G}$ has global finite F-representation type (GFFRT) and that $R$ provides a noncommutative resolution for $R^{p^r}$.