Chevalley Groups Research Articles

Isomorphisms and elementary equivalence of Chevalley groups over commutative rings

It is proved that two Chevalley groups with indecomposable root systems of rank over commutative rings (which contain in addition for the types , , , , and , and for the type ) are isomorphic or elementarily equivalent if and only if the corresponding root systems coincide, the weight lattices of the representation of the Lie algebra coincide, and the rings are isomorphic or elementarily equivalent, respectively. The isomorphisms of adjoint (elementary) Chevalley groups over the rings of the above types are also described. Bibliography: 25 titles.

On the values of unipotent characters of finite Chevalley groups of type E6 in characteristic 3

Let G be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of G by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on G. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where G is one of the groups E6(q), E62(q), and q is a power of the bad prime p=3 for E6, by exploiting known facts about the representation theory of the Hecke algebra associated with G.

Isomorphisms and elementary equivalence of Chevalley groups over commutative rings

В настоящей работе доказано, что две группы Шевалле с неразложимыми системами корней ранга $>1$ над коммутативными кольцами (содержащими дополнительно $1/2$ для типов $\mathbf A_2$, $\mathbf B_l$, $\mathbf C_l$, $\mathbf F_4$ и $\mathbf G_2$ и $1/3$ для типа $\mathbf G_2$) изоморфны или элементарно эквивалентны тогда и только тогда, когда соответствующие системы корней совпадают, решетки весов представления алгебры Ли совпадают, а кольца изоморфны или элементарно эквивалентны соответственно. Также описаны изоморфизмы присоединенных (элементарных) групп Шевалле над кольцами описанных типов. Библиография: 25 названий.

Supercharacter theories for Sylow p-subgroups of Lie type G 2

We construct a supercharacter theory, and establish the supercharacter table for Sylow p-subgroups of the Chevalley groups of Lie type when p>2. Then we calculate the conjugacy classes, determine the complex irreducible characters by Clifford theory, and obtain the character tables for when p>3.

On some distance-regular graphs with many vertices

We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $$G_2(4)$$ and $$G_2(5)$$, the orthogonal group O(7, 3) and the Tits group $$T=$$$$^2F_4(2)'$$. Most of the constructed graphs have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained graphs are new.

Commensurability growths of algebraic groups

Fixing a subgroup $$\varGamma $$ in a group G, the full commensurability growth function assigns to each n the cardinality of the set of subgroups $$\varDelta $$ of G with $$[\varGamma : \varGamma \cap \varDelta ][\varDelta : \varGamma \cap \varDelta ] \le n$$. For pairs $$\varGamma \le G$$, where G is a higher rank Chevalley group scheme defined over $${\mathbb {Z}}$$ and $$\varGamma $$ is an arithmetic lattice in G, we give precise estimates for the full commensurability growth, relating it to subgroup growth and a computable invariant that depends only on G.

The Highest Dimension of Commutative Subalgebras in Chevalley Algebras

Let L (K) denotes a Chevalley algebra with the root system over a field K. In 1945 A. I. Mal’cev investigated the problem of describing abelian subgroups of highest dimension in complex simple Lie groups. He solved this problem by transition to complex Lie algebras and by reduction to the problem of describing commutative subalgebras of highest dimension in the niltriangular subalgebra. Later these methods were modified and applied for the problem of describing large abelian subgroups in finite Chevalley groups. The main result of this article allows to calculate the highest dimension of commutative subalgebras in a Chevalley algebra L (K) over an arbitrary field

The completion of the $3$-modular character table of the Chevalley group $F_4(2)$ and its covering group

Using computational methods, we complete the determination of the $3$-modular character table of the Chevalley group $F_4(2)$ and its covering group.

The irreducible characters of the Sylow p-subgroups of the Chevalley groups D6(pf) and E6(pf)

We parametrize the set of irreducible characters of the Sylow p-subgroups of the Chevalley groups D6(q) and E6(q), for an arbitrary power q of any prime p. In particular, we establish that the parametrization is uniform for p≥3 in type D6 and for p≥5 in type E6, while the prime 2 in type D6 and the primes 2, 3 in type E6 yield character degrees of the form qm/pi which force a departure from the generic situations. Also for the first time in our analysis we see a family of irreducible characters of a classical group of degree qm/pi where i>1 which occurs in type D6.

Finite-Dimensional Pointed Hopf Algebras Over Finite Simple Groups of Lie Type IV. Unipotent Classes in Chevalley and Steinberg Groups

We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSLn(q) and PSp2n(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one class of involutions in PSUn(2m).

On root-involutions and root-subgroups of E6(K) for fields K of characteristic two

The purpose of this paper is to investigate the root-involutions and root-subgroups of the Chevalley group [Formula: see text] for fields [Formula: see text] of characteristic two. The approach we follow is elementary and self-contained depends on the notion of [Formula: see text]-sets which we have introduced in [Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type [Formula: see text] for fields of characteristic two, Beit. Algebra Geom. 58 (2017) 529–534]. The approach is elementary on the account that it consists of little more than naive linear algebra. It is remarkable to mention that Chevalley groups over fields of characteristic two have not much been researched. This work may contribute in this regard. This paper is divided into three main sections: the first section is a combinatorial section, the second section is on relations among [Formula: see text]-sets, the last one is on Lie algebra.

On the values of unipotent characters in bad characteristic

Let G(q) be a Chevalley group over a finite field \mathbb F_q . By Lusztig’s and Shoji’s work, the problem of computing the values of the unipotent characters of G(q) is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on G(q) . We show that this issue can be reduced to the case where q is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach.

On Lie algebras of type F4 and Chevalley groups F4(K), E6(K), and 2E6(K) for fields K of characteristic two

In this article, we give an elementary and self-contained approach to construct the Lie algebras of type over an arbitrary field K of characteristic two. The Lie algebras are represented as subalgebras of , where AK is a 27-dimensional vector space over K. The Lie algebras of type for fields K of characteristic two have been constructed by the authors, using the notion of M sets. Here we follow the same notion to give an easy and effective construction of the corresponding Chevalley groups , and . It is remarkable to mention that most of the available literature on Chevalley groups does not deal with fields of characteristic two. Hence, this work aims to contribute in this regard.

Commutator width of Chevalley groups over rings of stable rank 1

Abstract It is shown that each element of the elementary Chevalley group of rank greater than 2 over a ring of stable rank 1 can be expressed as a product of few commutators.

ON ABSTRACT HOMOMORPHISMS OF CHEVALLEY GROUPS OVER THE COORDINATE RINGS OF AFFINE CURVES

The goal of this paper is to establish a general rigidity statement for abstract representations of elementary subgroups of Chevalley groups of rank ≥ 2 over a class of commutative rings that includes the localizations of 1-generated rings and the coordinate rings of affine curves. Our main result implies, for example, that any finite-dimensional representation of SLn(ℤ[X]) (n ≥ 3) over an algebraically closed field of characteristic zero has a standard description, yielding thereby the first unconditional rigidity statement for finitely generated linear groups other than arithmetic groups/lattices.

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.

New uniform diameter bounds in pro-$p$ groups

We give newupper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay–Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of groups with an analytic structure over a pro-$p$ domain (with exponent depending on the dimension); Chevalley groups over a pro-$p$ domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups.

Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over [formula omitted

It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple Lie algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over Zp, extending the proof given by Clozel for the group Γ1(SL2(Zp)), the first congruence kernel of SL2(Zp) for primes p>2.

On Bouc's and Thévenaz's conjectures on B-groups

Bouc proposed the following conjecture: a finite group G is nilpotent if and only if its largest quotient B-group β(G) is nilpotent. And Thévenaz proposed another conjecture: a finite group G is solvable if and only if its largest quotient B-group β(G) is solvable. Bouc has proven that his conjecture holds when G is solvable. In this paper, we consider some cases when G is not solvable. Let S be a nonabelian simple group except the Chevalley groups An(q), Dn(q), E6(q), and An2(q), if the group G has exactly one composition factor isomorphic to S, then β(G) is not solvable, of course, is not nilpotent. That means we prove the conjectures in these cases.

On the Normalizer of a Unipotent Root Subgroup in a Chevalley Group

In the present paper, the normalizer of a unipotent short and long root subgroup in a Chevalley group over an arbitrary field is calculated in detail. Surely this result is known to specialists. However the author could not find a reference to it.