The Steinberg Tensor Product Theorem for General Linear Group Schemes in the Verlinde Category

Abstract

The Steinberg tensor product theorem is a fundamental result in the modular representation theory of reductive algebraic groups. It describes any finite-dimensional simple module of highest weight $\lambda$ over such a group as the tensor product of Frobenius twists of simple modules with highest weights the weights appearing in a $p$-adic decomposition of $\lambda$, thereby reducing the character problem to a a finite collection of weights. In recent years this theorem has been extended to various quasi-reductive supergroup schemes. In this paper, we prove the analogous result for the general linear group scheme $GL(X)$ for any object $X$ in the Verlinde category $\mathrm{Ver}_p$.