Irreducible Subgroups of Simple Algebraic Groups – A Survey

Donna Testerman
Cambridge University Press, 2019
Book chapter

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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Donna Testerman
Cambridge University Press, 2019
Book chapter

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https://infoscience.epfl.ch/record/268009?ln=en

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On irreducible subgroups of simple algebraic groups

Claude Marion, Donna Testerman

Let G be a simple 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 irreducible KG-module, which is p-restricted, tensor indecomposable and rational. Assume that the restriction of V to H is irreducible. In this paper, we study the triples (G, H, V) of this form when G is a classical group and H is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples (G, H, V) to the case where G is an orthogonal group, V is a spin module and H normalizes an orthogonal decomposition of the natural KG-module.

Springer Heidelberg2017

Irreducibility of disconnected subgroups of exceptional algebraic groups

Soumaïa Nadia Ghandour

This dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. The classification of the maximal closed connected subgroups of simple algebraic groups was carried out by E. B. Dynkin, G. M. Seitz and D. M. Testerman. Their analysis for the classical groups was based primarily on a striking result: if G is a simple algebraic group and ø : G → SL(V ) is a tensor indecomposable irreducible rational representation then, with specified exceptions, the image of G is maximal among closed connected subgroups of one of the classical groups SL(V), Sp(V ) or SO(V ). In the case of closed, not necessarily connected, subgroups of the classical groups, one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups: given a simple algebraic group Y defined over an algebraically closed field K, one is led to study the embeddings G < Aut(Y ), where G and Aut(Y ) are closed subgroups of SL(V ) and V is an irreducible rational KY -module on which G acts irreducibly. A partial analysis of such embeddings in the case of classical algebraic groups Y was carried out by B. Ford. We purpose to classify all triples (G, Y, V ) where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p > 0, G is a closed non-connected positive dimensional subgroup of Y and V is a nontrivial irreducible rational KY -module such that V|G is irreducible. We obtain a precise description of such triples (G, Y, V ).

EPFL2008

Irreducibility In Algebraic Groups And Regular Unipotent Elements

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We study (connected) reductive subgroups G of a reductive algebraic group H, where G contains a regular unipotent element of H. The main result states that G cannot lie in a proper parabolic subgroup of H. This result is new even in the classical case H = SL(n, F), the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jordan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group H which contain a regular unipotent element.

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