scholarly journals Relative subgroups in Chevalley groups

2010 ◽  
Vol 5 (3) ◽  
pp. 603-618 ◽  
Author(s):  
R. Hazrat ◽  
V. Petrov ◽  
N. Vavilov
Keyword(s):  
Commutative Ring ◽  
Chevalley Group ◽  
Congruence Subgroup ◽  
Admissible Pair ◽  
Adjoint Representation ◽  
Chevalley Groups ◽  
Main Structure ◽  
Elementary Group ◽  
Structure Theorems ◽  
Rank 2

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.

scholarly journals Generation of relative commutator subgroups in Chevalley groups. II

2020 ◽  
Vol 63 (2) ◽  
pp. 497-511 ◽  
Author(s):  
Nikolai Vavilov ◽  
Zuhong Zhang
Keyword(s):  
Root System ◽  
Commutative Ring ◽  
Chevalley Group ◽  
Congruence Subgroup ◽  
Unexpected Result ◽  
Chevalley Groups ◽  
Elementary Subgroup ◽  
Commutator Formula ◽  
Rank 2 ◽  
Relative Commutator

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).

Normality in Elementary Subgroups of Chevalley Groups Over Rings

1976 ◽  
Vol 28 (2) ◽  
pp. 420-428 ◽  
Author(s):  
James F. Hurley
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Chevalley Group ◽  
Complex Field ◽  
Normal Subgroups ◽  
Chevalley Groups ◽  
Simple Lie Algebra ◽  
Elementary Subgroup ◽  
Finite Dimensional

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

Normal Structure of Maximal Parabolic Subgroups in Chevalley Groups over Rings

2009 ◽  
Vol 16 (04) ◽  
pp. 631-648 ◽  
Author(s):  
Anastasia Stavrova
Keyword(s):  
Commutative Ring ◽  
Parabolic Subgroup ◽  
Chevalley Group ◽  
Normal Structure ◽  
Levi Subgroup ◽  
Parabolic Subgroups ◽  
Chevalley Groups ◽  
Maximal Parabolic Subgroup ◽  
Elementary Subgroup ◽  
Maximal Parabolic Subgroups

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.

Elements of Order Coxeter Number +1 in Chevalley Groups

1982 ◽  
Vol 34 (4) ◽  
pp. 945-951 ◽  
Author(s):  
Bomshik Chang
Keyword(s):  
Weyl Group ◽  
Chevalley Group ◽  
Trivial Solution ◽  
Affirmative Answer ◽  
Chevalley Groups ◽  
General Solutions ◽  
Coxeter Number ◽  
Image Position

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

scholarly journals Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

2014 ◽  
Vol 17 (1) ◽  
pp. 109-122 ◽  
Author(s):  
Simon M. Goodwin ◽  
Peter Mosch ◽  
Gerhard Röhrle
Keyword(s):  
Computer Program ◽  
Explicit Formula ◽  
Chevalley Group ◽  
Conjugacy Classes ◽  
Chevalley Groups ◽  
Sylow Subgroups ◽  
Finite Chevalley Group ◽  
Integer Coefficients ◽  
The Third

AbstractLet$G(q)$be a finite Chevalley group, where$q$is a power of a good prime$p$, and let$U(q)$be a Sylow$p$-subgroup of$G(q)$. Then a generalized version of a conjecture of Higman asserts that the number$k(U(q))$of conjugacy classes in$U(q)$is given by a polynomial in$q$with integer coefficients. In [S. M. Goodwin and G. Röhrle,J. Algebra321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of$k(U(q))$. By implementing it into a computer program using$\mathsf{GAP}$, they were able to calculate$k(U(q))$for$G$of rank at most five, thereby proving that for these cases$k(U(q))$is given by a polynomial in$q$. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of$k(U(q))$for finite Chevalley groups of rank six and seven, except$E_7$. We observe that$k(U(q))$is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write$k(U(q))$as a polynomial in$q-1$, then the coefficients are non-negative.Under the assumption that$k(U(q))$is a polynomial in$q-1$, we also give an explicit formula for the coefficients of$k(U(q))$of degrees zero, one and two.

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

2019 ◽  
Vol 18 (01) ◽  
pp. 1950017 ◽  
Author(s):  
S. Aldhafeeri ◽  
M. Bani-Ata
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Algebra ◽  
Chevalley Group ◽  
Chevalley Groups ◽  
Type Formula ◽  
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.

scholarly journals THE GEOMETRY OF BLUEPRINTS PART II: TITS–WEYL MODELS OF ALGEBRAIC GROUPS

2018 ◽  
Vol 6 ◽  
Author(s):  
OLIVER LORSCHEID
Keyword(s):  
Weyl Group ◽  
Chevalley Group ◽  
Tropical Geometry ◽  
Linear Representation ◽  
Unified Approach ◽  
Chevalley Groups ◽  
General Linear Groups ◽  
Base Extension ◽  
Field With One Element ◽  
Intrinsic Definition

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_{1}$, the field with one element. Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category$\text{Sch}_{{\mathcal{T}}}$ comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for ${\mathcal{G}}$ in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of ${\mathcal{G}}$ defines a model $G$ in $\text{Sch}_{{\mathcal{T}}}$ whose Weyl extension is the Weyl group $W$ of ${\mathcal{G}}$. We call such models Tits–Weyl models. The potential of Tits–Weyl models lies in (a) their intrinsic definition that is given by a linear representation; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.

scholarly journals Commutator width of Chevalley groups over rings of stable rank 1

2019 ◽  
Vol 22 (1) ◽  
pp. 83-101
Author(s):  
Andrei Smolensky
Keyword(s):  
Chevalley Group ◽  
Stable Rank ◽  
Chevalley Groups ◽  
Commutator Width ◽  
Elementary Chevalley Group

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.

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