Reducible subgroups of exceptional algebraic groups

Let G be a simple algebraic group over an algebraically closed field. A closed subgroup H of G is called G-completely reducible (G-cr) if, whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi factor of P. In this paper we complete the classification of connected G-cr subgroups when G has exceptional type, by determining the $L_0$-irreducible connected reductive subgroups for each simple classical factor $L_0$ of a Levi subgroup of G. As an illustration, we determine all reducible, G-cr semisimple subgroups when G has type $F_4$ and various properties thereof. This work complements results of Lawther, Liebeck, Seitz and Testerman, and is vital in classifying non-G-cr reductive subgroups, a project being undertaken by the authors elsewhere.     

Irreducible algebraic sets over divisible decomposed rigid groups

A soluble group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > …> G _p > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion-free when treated as right ℤ[G/G _i ]-modules. Free soluble groups are important examples of rigid groups. A rigid group G is divisible if elements of a quotient G _i /G _i+1 are divisible by nonzero elements of a ring ℤ[G/G _i ], or, in other words, G _i /G _i+1 is a vector space over a division ring Q(G/G _i ) of quotients of that ring. A rigid group G is decomposed if it splits into a semidirect product A ₁ A ₂…A _p of Abelian groups A _i ≅ G _i /G _i+1. A decomposed divisible rigid group is uniquely defined by cardinalities α _i of bases of suitable vector spaces A _i , and we denote it by M(α₁,…, α _p ). The concept of a rigid group appeared in [arXiv:0808.2932v1 [math.GR], ], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [11], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group M(α₁,…, α _p ). Our present goal is to derive important information directly about algebraic geometry over M(α₁,… α _p ). Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over M(α₁,…, α _p ) using the language of equations.     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

On arithmetic subgroups of simple algebraic groups

Arithmetic subgroups of simple isotropic algebraic groups are described as subgroups full of root elements.     

Irreducible representations of semisimple algebraic groups and supersolvable lattices

Let Λ be the cross section lattice of an irreducible representation of a semisimple algebraic group. Certain combinatorial properties of Λ are studied. Supersolvable Λ?s are determined in terms of Dynkin diagrams. The characteristic polynomial of a supersolvable Λ is computed.     

Strongly dense free subgroups of semisimple algebraic groups

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type     

Intersections of conjugacy classes and subgroups of algebraic groups

We show that if is a reductive group, then th roots of conjugacy classes are a finite union of conjugacy classes, and that if is an algebraic overgroup of , then the intersection of with a conjugacy class of is a finite union of -conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.

     

Determination of solvable algebraic groups by dense integral subgroups

Let G and G′ be triangularizable algebraic groups defined over the field Q of rational numbers, and let Γ ? G_Q and Γ′ ? G′_Q be dense subgroups of them containing integral subgroups of finite index. A study is made of the conditions under which a birational isomorphism of G and G′ follows from an abstract isomorphism of Γ and Γ′.     

Subnormal subgroups of the groups of rational points of reductive algebraic groups

We prove that for a reductive algebraic group over an infinite field the group of rational points does not contain any noncentral finitely generated normal subgroups.

     

Irreducible characters for algebraic groups in characteristic two (iii)

In this note, we determine the irreducible characters for the special linear group SL(6,K) and S L(7,K) over an algebraically closed field K of characteristic 2, by using the theorem of Xi Nanhua [7] and the MATLAB software.     

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups.     

On the modality of parabolic subgroups of linear algebraic groups

For a linear algebraic group G over an algebraically closed field k and a parabolic subgroup P of G the modality of P is defined to be the maximal number of parameters upon which a family of G-orbits on Lie P _u depends and it is denoted by mod P, where P _u is the unipotent radical of P. The principal aim of this note is a generalization of two basic “monotonicity” results from [19] to positive characteristic: (1) If Θ is a semisimple automorphism of G and P is Θ-stable, then mod P ^\Θ≤ mod P. (2) If G is reductive, char k is a good prime for G, and H is a closed reductive subgroup of G normalized by a maximal torus T⊂P of G, then mod (P∩H)≤ mod P. Received: 22 April 1998 / Revised version: 3 July 1998