Subgroups Of Chevalley Groups Research Articles

Finite index subgroups in Chevalley groups are bounded: An addendum to “On Bi-Invariant Word Metrics”

We prove that finite index subgroups in [Formula: see text]-arithmetic Chevalley groups are bounded.

Bruhat decomposition for carpet subgroups of Chevalley groups over fields

Bruhat decomposition for carpet subgroups of Chevalley groups over fields

Bruhat Decomposition for Carpet Subgroups of Chevalley Groups Over Fields

Bruhat Decomposition for Carpet Subgroups of Chevalley Groups Over Fields

Generation of relative commutator subgroups in Chevalley groups. II

AbstractIn the present paper, which is a direct sequel of our paper [14] joint with Roozbeh Hazrat, we prove an unrelativized version of the standard commutator formula in the setting of Chevalley groups. Namely, let Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let I,J be two ideals of R. We consider subgroups of the Chevalley group G(Φ, R) of type Φ over R. The unrelativized elementary subgroup E(Φ, I) of level I is generated (as a group) by the elementary unipotents xα(ξ), α ∈ Φ, ξ ∈ I, of level I. Obviously, in general, E(Φ, I) has no chance to be normal in E(Φ, R); its normal closure in the absolute elementary subgroup E(Φ, R) is denoted by E(Φ, R, I). The main results of [14] implied that the commutator [E(Φ, I), E(Φ, J)] is in fact normal in E(Φ, R). In the present paper we prove an unexpected result, that in fact [E(Φ, I), E(Φ, J)] = [E(Φ, R, I), E(Φ, R, J)]. It follows that the standard commutator formula also holds in the unrelativized form, namely [E(Φ, I), C(Φ, R, J)] = [E(Φ, I), E(Φ, J)], where C(Φ, R, I) is the full congruence subgroup of level I. In particular, E(Φ, I) is normal in C(Φ, R, I).

Reduction Theorems for Triples of Short Root Subgroups in Chevalley Groups

Reduction theorems for triples of short root unipotent subgroups in Chevalley groups of type B l and C l are proved. The main result asserts that except for one case, any subgroup generated by such a triple is conjugate to a subgroup of G(B4, K)U(B5, K) or G(C4, K)U(C5, K), respectively.

Generation of relative commutator subgroups in Chevalley groups

AbstractLet Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ∈ Φ ξ ∈ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ∈ Φ, ξ ∈ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ∈ Φ, ξ ∈ I, η ∈ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x_α(ζ)] and (iii) xα(ξζ), where α ∈ Φ, ξ ∈ I, ζ ∈ J, η ∈ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.

On the character degrees of Sylow p-subgroups of Chevalley groups G(pf ) of type E

Abstract Let 𝔽 q be a field of characteristic p with q elements. It is known that the degrees of the irreducible characters of the Sylow p-subgroup of GL(𝔽 q ) are powers of q. On the other hand Sangroniz (2003) showed that this is true for a Sylow p-subgroup of a classical group defined over 𝔽 q if and only if p is odd. For the classical groups of Lie type B, C and D the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow p-subgroups of the Chevalley groups D 4(q) with q = 2 f of degree q 3/2. Then we use an analogous construction for E 6(q) with q = 3 f to obtain characters of degree q 7/3, and for E 8(q) with q = 5 f to obtain characters of degree q 16/5. This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type E in terms of the representation theory of the Sylow p-subgroup.

Subring subgroups in Chevalley groups with doubly laced root systems

We study the lattice L of subgroups of a Chevalley group G(Φ,A) over a ring A, containing its elementary subgroup E(Φ,S) over a subring S⊆A. The standard description asserts that for any H∈L there exists a unique subring R⊇S of A such that H contains E(Φ,R) as a normal subgroup.The standard description was obtained by Ya. Nuzhin for algebraic field extensions. On the other hand, it was expected that the standard description does not hold for transcendental extensions. This was recently proved by the author for simply laced root systems.Now, suppose that Φ is doubly laced, i.e. Φ=Bl, Cl or F4, and that 2 is invertible in S. In the article we prove that in these settings the standard description of L holds for an arbitrary pair of rings S⊆A.

Relative subgroups in Chevalley groups

AbstractWe finish the proof of the main structure theorems for a Chevalley group G(Φ, R) of rank ≥ 2 over an arbitrary commutative ring R. Namely, we prove that for any admissible pair (A, B) in the sense of Abe, the corresponding relative elementary group E(Φ,R, A, B) and the full congruence subgroup C(Φ, R, A, B) are normal in G(Φ, R) itself, and not just normalised by the elementary group E(Φ, R) and that [E (Φ, R), C(Φ, R, A, B)] = E, (Φ, R, A, B). For the case Φ = F4 these results are new. The proof is new also for other cases, since we explicitly define C (Φ, R, A, B) by congruences in the adjoint representation of G (Φ, R) and give several equivalent characterisations of that group and use these characterisations in our proof.

Normal Structure of Maximal Parabolic Subgroups in Chevalley Groups over Rings

We study the normal structure of maximal parabolic subgroups of a Chevalley group over a commutative ring. More precisely, we describe the subgroups of a maximal parabolic subgroup P normalized by the elementary part of its Levi subgroup. As a corollary, we obtain a description of the subgroups in P normalized by its elementary subgroup EP.

Weakly closed unipotent subgroups in Chevalley groups

The goal of this note is to classify the weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that the Lie algebra of a connected simple algebraic group fails to be a so-called 2F-module.

Generation of pairs of short root subgroups in Chevalley groups

Generation of pairs of short root subgroups in Chevalley groups

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let G(k) be the Chevalley group of normal type associated with a root system G = Φ, or of twisted type G = mΦ,m = 2,3, over a field K. Its root subgroups Xs, for all possible s ∈ G+, generate a maximal unipotent subgroup U = UG(k) if p = charK < 0, U is a Sylow p-subgroup of G(K). We examine G and K for which there exists a paired intersection U ⋂ U9, g ∈ G(K), which is not conjugate in G(K) to a normal subgroup of U. If K is a finite field, this is equivalent to a condition that the normalizer of U ⋂ U9 in G(K)has a p-multiple index. Put p(Φ) = max(r,r)/(s,s) | r,s ∈ Φ. We prove a statement (Theorem 1) saying the following. Let G(K) be a Chevalley group of Lie rank greater than 1 over a finite field K of characteristic p and U be its Sylow p-subgroup equal to UG(K); also, either G = Φ and p(Φ) is distinct from p and 1, or G(K) is a twisted group. Then G(K) contains a monomial element n such that the normalizer U ⋂ of Un in G(K) has a p-multiple index. Let K be an associative commutative ring with unity and Φ(K,J) be a congruence subgroup of the Chevalley group Φ(K) modulo a nilpotent ideal J. We examine an hypercentral series 1 ⊂ Z1 ⊂ Z2 ⊂...⊂ Zc-1 of the group UΦ(K)⊂ (K,J). Theorem 2 shows that under an extra restriction on the quotient (Jt : J) of ideals, central series are related via Zi = Tc-iC, 1 ⩽ i < c, where C is a subgroup of central diagonal elements. Such a connection exists, in particular, if K = Zpm and J = (pd), 1 ⩽ d < m, d| m.

REPRESENTATIONS OF DEGREE TWO FOR ELEMENTARY SUBGROUPS OVER RINGS

We describe the representations of degree two for the elementary subgroups of Chevalley groups over a commutative ring that contains an infinite field. Those representations are found having a unique form which is characterized by a map between the ring and the base field of the representation space. The elementary subgroups admitting of a non-trivial representation of degree two are also distinguished. The representations concerned are abstract.

Generalized congruence subgroups of Chevalley groups

Generalized congruence subgroups of Chevalley groups

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rank n defined over the field k, where n⩾3. Let Φ be the root system associated with G(k) and let Π={α1, α2, …, αn} be a set of fundamental roots of Φ, with Φ+ being the set of positive roots of Φ with respect to Π. For α∈Π and γ∈Φ+, let nα(γ) be the coefficient of α in the expression of γ as a sum of fundamental roots; so γ=∑α∈Πnα(γ)α. Also we recall that ht(γ), the height of γ, is given by ht(γ)=∑α∈Πnα(γ). The highest root in Φ+ will be denoted by ρ. We additionally assume that the Dynkin diagram of G(k) is connected.

On simple periodic linear groups— Dense subgroups, permutation representations, and induced modules

We determine the Zariski-dense subgroups of Chevalley groups and their twisted analogues over infinite algebraic extensions of finite fields. It turns out that these are essentially forms of the same group (possibly becoming twisted) over smaller infinite fields. It follows from our classification that if\(\bar G\) is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of\(\bar G\) can never be maximal, and so the maximal subgroups of\(\bar G\) are necessarily closed. It follows that Seitz’s determination of the closed maximal subgroups of\(\bar G\) actually gives all the maximal subgroups.

Commutator structure of some subgroups of chevalley groups

Question 6.34 of the Kourov Notebook is solved: carry the familiar theorem of Merzlyakov on the joint commutator subgroup of covering subgroups over to Chevalley groups and moreover to orthogonal and unitary groups.

Rigid cyclic subgroups in Chevalley groups, I

Rigid cyclic subgroups in Chevalley groups, I

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system Φ. The parabolic subgroups of G are described in the work. A system σ=(σα) of ideals σα in R (α runs through all roots of the system Φ) is called a net of ideals in the commutative ring R if σασβ⊂σα+β for all those roots α and β for which α+β is also a root. A net σ is called parabolic if σα=R for α>0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets σ. The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.