55 research outputs found
Completely reducible SL(2)-homomorphisms
Let K be any field, and let G be a semisimple group over K. Suppose the
characteristic of K is positive and is very good for G. We describe all group
scheme homomorphisms phi:SL(2) --> G whose image is geometrically G-completely
reducible -- or G-cr -- in the sense of Serre; the description resembles that
of irreducible modules given by Steinberg's tensor product theorem. In case K
is algebraically closed and G is simple, the result proved here was previously
obtained by Liebeck and Seitz using different methods. A recent result shows
the Lie algebra of the image of phi to be geometrically G-cr; this plays an
important role in our proof.Comment: AMS LaTeX 20 page
On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups
Let be a vector space over an algebraically closed field . Let be
a quasisimple group of Lie type of characteristic acting
irreducibly on . Suppose also that is a classical group with natural
module , chosen minimally with respect to containing the image of under
the associated representation. We consider the question of when can act
irreducibly on a -constituent of and study its relationship
to the maximal subgroup problem for finite classical groups.Comment: To appear in Journal of Pure and Applied Algebr
Irreducible subgroups of algebraic groups
A closed subgroup of a semisimple algebraic group G is said to be Gâirreducible if it lies in no proper parabolic subgroup of G. We prove a number of results concerning such subgroups. Firstly they have only finitely many overgroups in G; secondly, with some specified exceptions, there exist Gâirreducible subgroups of type A1; and thirdly, we prove an embedding theorem for Gâirreducible subgroup
On irreducible subgroups of simple algebraic groups
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
Centres of centralizers of unipotent elements in simple algebraic groups
Let G be a simple algebraic group defined over an algebraically closed field k whose characteristic is either 0 or a good prime for G, and let u is an element of G be unipotent. We study the centralizer C-G(u), especially its centre Z(C-G(u)). We calculate the Lie algebra of Z(C-G(u)), in particular determining its dimension; we prove a succession of theorems of increasing generality, the last of which provides a formula for dim Z(C-G(u)) in terms of the labelled diagram associated to the conjugacy class containing u
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