## Irreducible Subgroups of Simple Algebraic Groups – A Survey

Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p≥ 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.

Editor(s):
Campbell, C. M.
Parker, C. W.
Quick, M. R.
Robertson, E. F.
Roney-Dougal, C. M.
Published in:
Groups St Andrews 2017 in Birmingham, 230-260
Year:
2019-04
Publisher:
Cambridge, Cambridge University Press
ISBN:
9781108692397
Laboratories: