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

TL;DR

This paper investigates conditions under which tensor products of simple modules for algebraic groups are multiplicity free or completely reducible, providing complete classifications for groups SL_3 and Sp_4 over algebraically closed fields.

Contribution

It develops general tools to determine multiplicity freeness and complete reducibility of tensor products and applies them to classify these properties for SL_3 and Sp_4.

Findings

Complete classification for SL_3 tensor products

Complete classification for Sp_4 tensor products

Development of general analytical tools

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 $G = \mathrm{Sp}_4(\Bbbk)$.