118 research outputs found
On irreducible subgroups of simple algebraic groups
Get PDFLet 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
Reductive overgroups of distinguished unipotent elements in simple algebraic groups
Get PDFLet be a simple linear algebraic group over an algebraically closed field of characteristic . In this thesis, we investigate closed connected reductive subgroups that contain a given distinguished unipotent element of . Our main result is the classification of all such that are maximal among the closed connected subgroups of .
When is simple of exceptional type, the result is easily read from the tables computed by Lawther (J. Algebra, 2009). Our focus is then on the case where is simple of classical type, say , , or . We begin by considering the maximal closed connected subgroups of which belong to one of the families of the so-called \emph{geometric subgroups}. Here the only difficult case is the one where is the stabilizer of a tensor decomposition of . For and , we solve the problem with explicit calculations; for the other tensor product subgroups we apply a result of Barry (Comm. Algebra, 2015).
After the geometric subgroups, the maximal closed connected subgroups that remain are the such that is simple and is an irreducible and tensor indecomposable -module. The bulk of this thesis is concerned with this case. We determine all triples where is a simple algebraic group, is a unipotent element, and is a rational irreducible representation such that is a distinguished unipotent element of . When , this was done in previous work by Liebeck, Seitz and Testerman (Pac. J. Math, 2015).
In the final chapter of the thesis, we consider the more general problem of finding all connected reductive subgroups of that contain a distinguished unipotent element of . This leads us to consider connected reductive overgroups of which are contained in some proper parabolic subgroup of . Testerman and Zalesski (Proc. Am. Math. Soc, 2013) have shown that when is a regular unipotent element of , no such exists. We give several examples which show that their result does not generalize to distinguished unipotent elements. As an extension of the Testerman-Zalesski result, we show that except for two known examples which occur in the case where , a connected reductive overgroup of a distinguished unipotent element of order cannot be contained in a proper parabolic subgroup of .GR-TE
Unipotent Overgroups in Simple Algebraic GroupsGroups of Exceptional Type, Coxeter Groups and Related Geometries
No full textIrreducible almost simple subgroups of classical algebraic groups
No full textLet G be a simple classical algebraic group over an algebraically closed field K of characteristic p ≥ 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable p-restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V ) of this form, where H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension
Irreducible almost simple subgroups of classical algebraic groups
Get PDFLet G be a simple classical algebraic group over an algebraically closed field K of characteristic p ≥ 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial p-restricted irreducible tensor indecomposable rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G, H, V ) of this form, where V ≠ W, W∗ and H is a disconnected almost simple positive-dimensional closed subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible triples (G, H, V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension
Irreducibility of disconnected subgroups of exceptional algebraic groups
No full textThis 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 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 ).GR-TE
MULTIPLICITY-FREE REPRESENTATIONS OF THE PRINCIPAL A1-SUBGROUP IN A SIMPLE ALGEBRAIC GROUP
No full textLet G be a simple algebraic group defined over an algebraically closed field k of characteristic p > 0. For p ≥ h, the Coxeter number of G, any regular unipotent element of G lies in an A1-subgroup of G; there is a unique G-conjugacy class of such subgroups and any member of this class is a so-called “principal A1-subgroup of G”. Here we classify all irreducible k G-modules whose restriction to a principal A1-subgroup of G has no repeated composition factors, extending the work of Liebeck, Seitz and Testerman which treated the same question when k is replaced by an algebraically closed field of characteristic zero.GR-TE
A<sub>1 </sub>-type subgroups containing regular unipotent elements
Get PDFLet G be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic p > 0 and let X = PSL2(p) be a subgroup of G containing a regular unipotent element x of G . By a theorem of Testerman, x is contained in a connected subgroup of G of type A1 . In this paper we prove that with two exceptions, X itself is contained in such a subgroup (the exceptions arise when (G, p) = (E6, 13) or (E7, 19) ). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on p and the embedding of X in G . We discuss applications of our main result to the study of the subgroup structure of finite groups of Lie type
Irreducible subgroups of simple algebraic groups - a survey
Get PDFLet 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
Irreducible subgroups of simple algebraic groups - a survey
Get PDFLet 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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