Multiplicity free and completely reducible tensor products for $\mathrm{SL}_3(\Bbbk)$ and $\mathrm{Sp}_4(\Bbbk)$

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $\Bbbk$ of positive characteristic. We consider the questions of when the tensor product of two simple $G$-modules is multiplicity free or completely reducible. We develop tools for answering these questions in general, and we use them to provide complete answers for the groups $G = \mathrm{SL}_3(\Bbbk)$ and $G = \mathrm{Sp}_4(\Bbbk)$.