The Frobenius morphism in invariant theory

Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=\operatorname{Gr}(2,n)$ defined over an algebraically closed field of characteristic $p>0$. In this paper we give a completely characteristic free description of the decomposition of $R$, considered as a graded $R^p$-module, into indecomposables ("Frobenius summands"). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of $\mathbb{G}$ and we obtain in particular that this pushforward is almost never a tilting bundle. On the other hand we show that $R$ provides a "noncommutative resolution" for $R^p$ when $p\ge n-2$, generalizing a result known to be true for toric varieties. In both the invariant theory and the geometric setting we observe that if the characteristic is not too small the Frobenius summands do not depend on the characteristic in a suitable sense. In the geometric setting this is an explicit version of a general result by Bezrukavnikov and Mirkovi\'c on Frobenius decompositions for partial flag varieities. We are hopeful that it is an instance of a more general "$p$-uniformity" principle.