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).

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.

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.

scholarly journals On the characters of the Sylow -subgroups of untwisted Chevalley groups

2016 ◽  
Vol 19 (2) ◽  
pp. 303-359 ◽  
Author(s):  
Frank Himstedt ◽  
Tung Le ◽  
Kay Magaard
Keyword(s):  
Root System ◽  
Chevalley Group ◽  
Positive Root ◽  
Chevalley Groups ◽  
Irreducible Characters ◽  
Single Root ◽  
The Family ◽  
Positive Roots ◽  
Root Character

Let$UY_{n}(q)$be a Sylow$p$-subgroup of an untwisted Chevalley group$Y_{n}(q)$of rank$n$defined over $\mathbb{F}_{q}$where$q$is a power of a prime$p$. We partition the set$\text{Irr}(UY_{n}(q))$of irreducible characters of$UY_{n}(q)$into families indexed by antichains of positive roots of the root system of type$Y_{n}$. We focus our attention on the families of characters of$UY_{n}(q)$which are indexed by antichains of length$1$. Then for each positive root$\unicode[STIX]{x1D6FC}$we establish a one-to-one correspondence between the minimal degree members of the family indexed by$\unicode[STIX]{x1D6FC}$and the linear characters of a certain subquotient$\overline{T}_{\unicode[STIX]{x1D6FC}}$of$UY_{n}(q)$. For$Y_{n}=A_{n}$our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of$\text{Irr}(UE_{i}(q))$,$6\leqslant i\leqslant 8$, and$\text{Irr}(UF_{4}(q))$.

scholarly journals Generation of relative commutator subgroups in Chevalley groups

2015 ◽  
Vol 59 (2) ◽  
pp. 393-410 ◽  
Author(s):  
R. Hazrat ◽  
N. Vavilov ◽  
Z. Zhang
Keyword(s):  
Normal Subgroup ◽  
Chevalley Group ◽  
Commutator Subgroup ◽  
Normal Subgroups ◽  
Elementary Subgroup ◽  
Elementary Group ◽  
Commutator Width ◽  
Relative Commutator ◽  
Elementary Generator ◽  
Elementary Chevalley Group

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.

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 Lefschetz Operators, Hodge–Riemann Forms, and Representations

2020 ◽  
Author(s):  
Peter Fiebig
Keyword(s):  
Root System ◽  
Bilinear Form ◽  
Chevalley Group ◽  
Simple Root ◽  
Complex Geometry ◽  
Linear Operators ◽  
Vector Spaces ◽  
Tilting Module ◽  
Weight Lattice ◽  
Simply Connected

Abstract For a field of characteristic $\ne 2$, we study vector spaces that are graded by the weight lattice of a root system and are endowed with linear operators in each simple root direction. We show that these data extend to a weight lattice graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.

scholarly journals The Chevalley group G2(2) of order 12096 and the octonionic root system of E7

2007 ◽  
Vol 422 (2-3) ◽  
pp. 808-823 ◽  
Author(s):  
Mehmet Koca ◽  
Ramazan Koç ◽  
Nazife Ö. Koca
Keyword(s):  
Root System ◽  
Chevalley Group

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.

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