Irreducible subgroups of simple algebraic groups - a survey
Timothy C. Burness, Donna M. Testerman

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 be a simple linear algebraic group over an algebraically closed field of characteristic , let be a proper closed subgroup of and let be a nontrivial finite dimensional irreducible rational -module. We say that is an irreducible triple if is irreducible as a -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.
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