Subsystem Subgroups Research Articles

Overgroups of subsystem subgroups in exceptional groups: inside a sandwich

This paper supplements a previous paper by the author (Overgroups of subsystem subgroups in exceptional groups: a 2 A 1 2A_1 -proof, (2020)), where the overgroup lattice of the elementary subsystem subgroup E ( Δ , R ) E(\Delta ,R) of the Chevalley group G ( Φ , R ) G(\Phi ,R) for a sufficiently large root subsystem Δ \Delta was studied. Currently, the subject-matter is the relationship between the elementary subgroup E ^ ( σ ) \widehat {E}(\sigma ) given by a net of ideals σ \sigma of the ring R R , and the stabilizer S ( σ ) S(\sigma ) of the corresponding Lie subalgebra of the Chevalley algebra. In particular, it is proved that under a certain condition the subgroup E ^ ( σ ) \widehat {E}(\sigma ) is normal in S ( σ ) S(\sigma ) , and some properties of the corresponding quotient group are established.

Special factors in restrictions of irreducible modules of special linear and symplectic groups to subsystem subgroups with two simple components

The restrictions of irreducible modules of special linear and symplectic groups in an odd characteristic p with p-large highest weights to a subsystem subgroup H of maximal rank with two simple components H1 and H2 are considered. The lower estimate for the number of composition factors for such restrictions, which are p-large for the subgroup H1 and are not too small for H2, is found. The lower estimates of the number of Jordan blocks of maximal size for the images of certain unipotent elements in the corresponding representations of such groups are determined.

Overgroups of subsystem subgroups in exceptional groups: nonideal levels

In the present paper, a description of overgroups for the subsystem subgroups E ( Δ , R ) E(\Delta ,R) of the Chevalley groups G ( Φ , R ) G(\Phi ,R) over the ring R R , where Φ \Phi is a simply laced root system and Δ \Delta is its sufficiently large subsystem, is almost entirely finished. Namely, objects called levels are defined and it is shown that for any such overgroup H H there exists a unique level σ \sigma with E ( σ ) ≤ H ≤ Stab G ( Φ , R ) ⁡ ( L max ( σ ) ) E(\sigma )\le H\le \operatorname {Stab}_{G(\Phi ,R)}(L_{\max }(\sigma )) , where E ( σ ) E(\sigma ) is an elementary subgroup associated with the level σ \sigma and L max ( σ ) L_{\max }(\sigma ) is the corresponding subalgebra of the Chevalley algebra. Unlike the previous papers, here levels can be more complicated than nets of ideals.

Overgroups of subsystem subgroups in exceptional groups: A 2𝐴₁-proof

In the present paper, a weak form of sandwich classification for the overgroups of the subsystem subgroup E ( Δ , R ) E(\Delta ,R) of the Chevalley group G ( Φ , R ) G(\Phi ,R) is proved in the case where Φ \Phi is a simply laced root system and Δ \Delta is its sufficiently large subsystem. Namely, it is shown that, for such an overgroup H H , there exists a unique net of ideals σ \sigma of the ring R R such that E ( Φ , Δ , R , σ ) ≤ H ≤ Stab G ( Φ , R ) ⁡ ( L ( σ ) ) E(\Phi ,\Delta ,R,\sigma )\le H\le \operatorname {Stab}_{G(\Phi ,R)}(L(\sigma )) , where E ( Φ , Δ , R , σ ) E(\Phi ,\Delta ,R,\sigma ) is an elementary subgroup associated with the net and L ( σ ) L(\sigma ) is the corresponding subalgebra of the Chevalley Lie algebra.

Improved K1-stability for the embedding D5 into E6

This article is dedicated to the surjective stability of the K1-functor for Chevalley groups for the embedding D5 into E6. This case was already studied by Plotkin. In this article, we improve his result by showing that surjective stability holds under a weaker assumption on a ring. Another result of this article shows how the K1-stability can help to study overgroups of subsystem subgroups.

The Block Structure of the Images of Regular Unipotent Elements from Subsystem Symplectic Subgroups of Rank 2 in Irreducible Representations of Symplectic Groups. I

The dimensions of the Jordan blocks in the images of regular unipotent elements from subsystem subgroups of type C2 in p-restricted irreducible representations of groups of type Cn in characteristic p ≥ 11 with locally small highest weights are found. These results can be applied for investigating the behavior of unipotent elements in modular representations of simple algebraic groups and recognizing representations and linear groups. The article consists of 3 parts. In the first one, preliminary lemmas that are necessary for proving the principal results, are contained and the case where all weights of the restriction of a representation considered to a subgroup of type A1 containing a relevant unipotent element are less than p, is investigated.

Subsystem subgroups of the group of type $\mathrm {F}_4$ generated by short root subgroups

Subsystem subgroups of the group of type $\mathrm {F}_4$ generated by short root subgroups

Decompositions of congruence subgroups of Chevalley groups

We formulate and prove relative versions of several decompositions known in the theory of Chevalley groups over commutative rings. These decompositions are used to obtain factorizations in terms of subsystem subgroups of type [Formula: see text] and upper estimates of the width of principal congruence subgroups with respect to Tits–Vaserstein generators.

The Restrictions of Representations of the Special Linear Group to Subsystem Subgroups of Type A2

The restrictions of irreducible representations of the special linear group over an algebraically closed field of positive characteristic p to subsystem subgroups of type A2 are studied. The composition factors for such restrictions are described in the case where the highest weights of the representations are locally small.

Regular Unipotent Elements from Subsystem Subgroups of Type A 2 in Representations of the Special Linear Groups

For p > 2, odd Jordan block sizes of the images of regular unipotent elements from subsystem subgroups of type A2 in irreducible p-restricted representations for groups of type Ar over the field of characteristic p, the weights of which are locally small with respect to p, are found. The weight is called locally small if the double sum of its two neighboring coefficients is less than p. This result is part of a more general program investigating the behavior of unipotent elements in representations of the classical algebraic groups. It can be used to solve recognition problems for representations or linear groups by the presence of certain elements.

Overgroups of Subsystem Subgroups in Exceptional Groups: Levels

An embedding of root systems ∆ ⊆ Φ determines the corresponding regular embedding G(∆, R)≤ G(Φ, R) of Chevalley groups, over an arbitrary commutative ring R. Denote by E(∆, R) the elementary subgroup of G(∆, R). In the present paper we initiate the study of intermediate subgroups H, E∆, R) ≤ H ≤ G(Φ, R), provided that Φ=E6, E7, E8, F4 or G2, and there are no roots in Φ orthogonal to all of ∆. There are 72 such pairs (Φ, Δ)$. For F4 and G2 we assume, moreover, that 2 ∈ R * or 6 ∈ R *, respectively. For all such subsystems Δ we construct the levels of intermediate subgroups. We prove that these levels are determined by certain systems of ideals in R, one for each Δ-equivalence class of roots in Φ\∆, and calculate all relations among these ideals, in each case. Bibliography: 64 titles.

Some good-filtration subgroups of simple algebraic groups

Let G be a connected and reductive algebraic group over an algebraically closed field of characteristic p>0. An interesting class of representations of G consists of those G-modules having a good filtration — i.e. a filtration whose layers are the induced highest weight modules obtained as the space of global sections of G-linearized line bundles on the flag variety of G. Let H⊂G be a connected and reductive subgroup of G. One says that (G,H) is a Donkin pair, or that H is a good filtration subgroup of G, if whenever the G-module V has a good filtration, the H-module resHGV has a good filtration.In this paper, we show when G is a “classical group” that the optimalSL2-subgroups of G are good filtration subgroups. We also consider the cases of subsystem subgroups in all types and determine some primes for which they are good filtration subgroups.

RESTRICTIONS OF MODULES OVER CLASSICAL GROUPS TO SUBGROUPS OF TYPE A1 × A1 IN CHARACTERISTIC 2

The restrictions of irreducible modules over classical groups in characteristic 2 to subsystem subgroups of type A1 × A1 are studied. The composition factors (without their multiplicities) for restrictions of irreducible modules over Ar(K) and Dr(K) with 2-restricted highest weights are found.

On the block structure of regular unipotent elements from subsystem subgroups of type A 1 × A 2 in representations of the special linear group

On the block structure of regular unipotent elements from subsystem subgroups of type A 1 × A 2 in representations of the special linear group

Restrictions of modules over classical groups to subgroups of type A 2 in characteristic 2

Restrictions of irreducible modules over classical groups to subsystem subgroups of type A 2 in characteristic 2 are studied. The composition factors (without their multiplicities) for restrictions of such modules with 2-restricted highest weights are found. Bibliography: 7 titles.

Restrictions of Modular Representations of Special Linear Groups to A1×A1 Subgroups

We study the restrictions of infinitesimally irreducible modular representations of algebraic groups of type An to subsystem subgroups of type A1 × A1. Under some restrictions on the highest weight of a representation we find their composition factors.

Regular unipotent elements from subsystem subgroups of rank 2 in modular representations of classical groups

Images of regular unipotent elements from subsystem subgroups of type A2 and B2 in irreducible modular representations of classical groups are studied. For images of such elements and representations with locally small highest weights, all sizes of Jordan blocks of one and the same parity are found. Bibliography: 17 titles.

Subgroups of symplectic groups that contain a subsystem subgroup

Let Γ = GSp(2l, R) be the general symplectic group of rank l over a commutative ring R such that 2 ∈ R*; and let ν be a symmetric equivalence relation on the index set {1,…, l, −l,…, 1} all of whose classes contain at least 3 elements. In the present paper, we prove that if a subgroup H of Γ contains the group EΓ(ν) of elementary block diagonal matrices of type ν, then H normalizes the subgroup generated by all elementary symplectic transvections Tij(ξ) ∈ H. Combined with the previous results, this completely describes the overgroups of subsystem subgroups in this case. Similar results for subgroups of GL(n, R) were established by Z. I. Borewicz and the author in the early 1980s, while for GSp(2l, R) and GO(n, R) they have been announced by the author in the late 1980s, but a complete proof for the symplectic case has not been published before. Bibliography: 74 titles.

Normalizers of subsystem subgroups in finite groups of Lie type

Finite groups of Lie type form the greater part of known finite simple groups. An important class of subgroups of finite groups of Lie type are so-called reductive subgroups of maximal rank. These arise naturally as Levi factors of parabolic groups and as centralizers of semisimple elements, and also as subgroups with maximal tori. Moreover, reductive groups of maximal rank play an important part in inductive studies of subgroup structure of finite groups of Lie type. Yet a number of vital questions dealing in the internal structure of such subgroups are still not settled. In particular, we know which quasisimple groups may appear as central multipliers in the semisimple part of any reductive group of maximal rank, but we do not know how normalizers of those quasisimple groups are structured. The present paper is devoted to tackling this problem.