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.

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.

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.

scholarly journals PARABOLIC GROUPS ACTING ON ONE-DIMENSIONAL COMPACT SPACES

2005 ◽  
Vol 15 (05n06) ◽  
pp. 893-906
Author(s):  
FRANÇOIS DAHMANI
Keyword(s):  
Hyperbolic Group ◽  
Surface Groups ◽  
Parabolic Subgroups ◽  
Maximal Parabolic Subgroup ◽  
One Dimensional ◽  
Compact Spaces ◽  
Dimensional Boundary ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic ◽  
Maximal Parabolic Subgroups

Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of one-dimensional connected boundaries. We get that any non-torsion infinite finitely-generated group is a maximal parabolic subgroup of some relatively hyperbolic group with connected one-dimensional boundary without global cut point. For boundaries homeomorphic to a Sierpinski carpet or a 2-sphere, the only maximal parabolic subgroups allowed are virtual surface groups (hyperbolic, or virtually ℤ + ℤ).

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals PREGARSIDE MONOIDS AND GROUPS, PARABOLICITY, AMALGAMATION, AND FC PROPERTY

2013 ◽  
Vol 23 (06) ◽  
pp. 1431-1467
Author(s):  
EDDY GODELLE ◽  
LUIS PARIS
Keyword(s):  
Parabolic Subgroup ◽  
Parabolic Subgroups ◽  
Basic Study ◽  
The Family ◽  
Definition Of ◽  
Amalgamated Product

We define the notion of preGarside group slightly lightening the definition of Garside group so that all Artin–Tits groups are preGarside groups. This paper intends to give a first basic study on these groups. Firstly, we introduce the notion of parabolic subgroup, we prove that any preGarside group has a (partial) complemented presentation, and we characterize the parabolic subgroups in terms of these presentations. Afterwards we prove that the amalgamated product of two preGarside groups along a common parabolic subgroup is again a preGarside group. This enables us to define the family of preGarside groups of FC type as the smallest family of preGarside groups that contains the Garside groups and that is closed by amalgamation along parabolic subgroups. Finally, we make an algebraic and combinatorial study on FC type preGarside groups and their parabolic subgroups.

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 Tight Quotients and Double Quotients in the Bruhat Order

10.37236/1871 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
John R. Stembridge
Keyword(s):  
Coxeter Groups ◽  
Upper Bounds ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Order Dimension ◽  
Parabolic Subgroups ◽  
Affine Weyl Groups ◽  
Positive Side ◽  
Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

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