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Irreducible Subgroups of Simple Algebraic Groups – A Survey

Published online by Cambridge University Press:  15 April 2019

C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
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Summary

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.

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Publisher: Cambridge University Press
Print publication year: 2019

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