  Let $G$ be a connected reductive group over an algebraically closed field of
characteristic $p>0$. Given an indecomposable G-module $M$, one can ask when it
remains indecomposable upon restriction to the Frobenius kernel $G_r$, and when
its $G_r$-socle is simple (the latter being a strictly stronger condition than
the former). In this paper, we investigate these questions for $G$ having an
irreducible root system of type A. Using Schur functors and inverse Schur
functors as our primary tools, we develop new methods of attacking these
problems, and in the process obtain new results about classes of Weyl modules,
induced modules, and tilting modules that remain indecomposable over $G_r$.
