Multiplicity-free tensor products of irreducible modules over simple algebraic groups in positive characteristic

TL;DR

This thesis classifies when tensor products of simple modules over certain algebraic groups are multiplicity-free, focusing on groups like SL_2, SL_3, and partial results for Sp_4 in positive characteristic.

Contribution

It provides a complete classification for SL_2 and SL_3, and partial results for SL_n with p=2 and Sp_4, advancing understanding of module tensor products in positive characteristic.

Findings

Classified multiplicity-free tensor products for SL_2 and SL_3.

Provided partial classifications for SL_n when p=2.

Extended results to Sp_4 in positive characteristic.

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. In this master thesis, we classify multiplicity-free tensor products of simple modules for the groups $SL_2(k)$ and $SL_3(k)$. We also provide a classification for $SL_n(k)$ when $p=2$ and give partial results in the case of $Sp_4(k)$.