The reductive subgroups of G 2

Stewart, David I.

doi:10.1515/jgt.2009.035

Cited by 11 publications

(15 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The only subgroups of G 2 that have the same composition factors as X are a LeviĀ 1 and A 1 < A 2 embedded via W (2). However, [28,Theorem 1] shows that X is not conjugate to either of these subgroups. Therefore X is not contained reducibly in another reductive, maximal connected subgroup nor is it contained in a Levi subgroup of G. Hence X satisfies the hypothesis of Corollary 5 and is listed in Table 2.…”

Section: Corollaries 3 and 4 Follow From Careful Consideration Of Thementioning

confidence: 99%

Irreducible A1 subgroups of exceptional algebraic groups

Thomas¹

2016

Journal of Algebra

View full text Add to dashboard Cite

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible A 1 subgroups of exceptional algebraic groups G. Consequences are given concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible A 1 subgroups are determined by their composition factors on the adjoint module of G.

show abstract

Section: Corollaries 3 and 4 Follow From Careful Consideration Of Thementioning

confidence: 99%

Irreducible A1 subgroups of exceptional algebraic groups

Thomas¹

2016

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…In [Ste10a] and [Ste12] we find all semisimple non-G-cr subgroups of G where G is G 2 and F 4 respectively. So the result follows for these cases.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Non-G-Completely Reducible Subgroups of the Exceptional Algebraic Groups

Stewart

2013

International Mathematics Research Notices

Self Cite

View full text Add to dashboard Cite

Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p > 0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a parabolic subgroup P of G, then H is in a Levi subgroup of that parabolic. Building on work of Liebeck and Seitz, we find all triples (X, G, p) such that there exists a closed, connected, simple non-G-cr subgroup H ≤ G with root system X.

show abstract

“…The idea is that when p = 0 there is no distinction between demanding that a subgroup H of a reductive group G is reductive or linearly reductive or G-completely reducible, but there is a huge difference in positive characteristic. The notion of complete reducibility was introduced by Serre [52] and Richardson 1 [49], and over the past twenty years or so has found many applications in the theory of algebraic groups, their subgroup structure and representation theory, geometric invariant theory, and the theory of buildings: for examples, see [2], [5], [9], [30], [35], [36], [39], [56], [57], [58].…”

Section: Introductionmentioning

confidence: 99%

Orbit closures and invariants

2019

View full text Add to dashboard Cite

Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let X H denote the set of fixed points of H in X, and N G (H) the normalizer of H in G. In this paper we study the natural map of quotient varieties ψ X,H : X H /N G (H) → X/G induced by the inclusion X H ⊆ X. We show that, given G and H, ψ X,H is a finite morphism for all affine G-varieties X if and only if H is a G-completely reducible subgroup of G (in the sense defined by J-P. Serre); this was proved in characteristic 0 by Luna in the 1970s. We discuss some applications and give a criterion for ψ X,H to be an isomorphism. We show how to extend some other results in Luna's paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic g ∈ G if and only if H ∩ gKg −1 is reductive for generic g ∈ G.

show abstract

The reductive subgroups of G 2

Cited by 11 publications

References 11 publications

Irreducible A1 subgroups of exceptional algebraic groups

Irreducible A1 subgroups of exceptional algebraic groups

Non-G-Completely Reducible Subgroups of the Exceptional Algebraic Groups

Orbit closures and invariants

Contact Info

Product

Resources

About