Irreducible subgroups of simple algebraic groups - a survey

TL;DR

This survey reviews recent progress in classifying irreducible triples, which are fundamental in understanding subgroup structures and representations of simple algebraic groups over algebraically closed fields.

Contribution

It compiles and discusses recent advances in classifying irreducible triples for positive dimensional subgroups of simple algebraic groups.

Findings

Progress in classifying irreducible triples for algebraic groups

Extension of earlier work to positive characteristic fields

Enhanced understanding of subgroup structures in simple algebraic groups

Abstract

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is irreducible as a $KH$-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.