Title: The Frobenius morphism in invariant theory,
Abstract: 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 Mirković on
Frobenius decompositions for partial flag varieities. We are hopeful that it is
an instance of a more general "$p$-uniformity" principle.