This dissertation investigates the socle of the Steinberg squares for an algebraic group in positive characteristic, and what this socle says about the decomposition of the Steinberg square into a direct sum of indecomposable modules.

A main tool in this investigation will be the fact that tensoring the Steinberg module with a simple module of restricted highest weight gives a module with a good filtration. This result was first proved by Andersen when the characteristic is large enough. In this dissertation, generalizations of those results, which are joint work with Daniel Nakano, are presented.

The main results of the dissertation provide formulas which describe how to find the multiplicities of simple modules in the socle of a Steinberg square, given information about the multiplicities of simple modules in Weyl modules. Further, it is shown that when the prime is large enough, the socle completely determines how a Steinberg square decomposes.

The dissertation also investigates the socle of the Steinberg square for a finite group of Lie type, again providing formulas which describe how to find the multiplicity of a simple module in the socle, given information about the multiplicities of simple modules in Weyl modules.