Steinberg squares and tensor products of tilting modules with simple modules

Abstract

Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\mathbb{k}$ of characteristic $p>0$. We establish an isomorphism of $G$-modules between a direct sum of modules $\text{St}\otimes \text{St}$ and a direct sum of tensor products of simple modules of restricted highest weight with tilting modules that are projective over the Frobenius kernel of $G$. This isomorphism holds precisely when Donkin's Tilting Module Conjecture does, and thus can be seen as providing a $G$-module theoretic characterization of this conjecture.

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