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

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb{k}$ 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 $G={\mathrm{SL}}_3(\mathbb{k})$ and $G={\mathrm{Sp}}_4(\mathbb{k})$.

arXiv commentreferences added, 49 pages, comments welcome