892 research outputs found
Normalizer of the Chevalley group of type
We consider the simply connected Chevalley group of type in
the 56-dimensional representation. The main objective of the paper is to prove
that the following four groups coincide: the normalizer of the elementary
Chevalley group , the normalizer of the Chevalley group
itself, the transporter of into , and the extended
Chevalley group . This holds over an arbitrary commutative
ring , with all normalizers and transporters being calculated in .
Moreover, we characterize as the stabilizer of a system of
quadrics. This last result is classically known over algebraically closed
fields, here we prove that the corresponding group scheme is smooth over
, which implies that it holds over arbitrary commutative rings.
These results are one of the key steps in our subsequent paper, dedicated to
the overgroups of exceptional groups in minimal representations.Comment: 23 pages, will be published in St. Petersburg Mathematical Journa
Generation of relative commutator subgroups in Chevalley groups. II
In the present paper, which is a direct sequel of our paper [12] joint with
Roozbeh Hazrat, we prove unrelativised version of the standard commutator
formula in the setting of Chevalley groups. Namely, let be a reduced
irreducible root system of rank , let be a commutative ring and let
be two ideals of . We consider subgroups of the Chevalley group
of type over . The unrelativised elementary subgroup
of level is generated (as a group) by the elementary unipotents
, , , of level . Obviously, in
general has no chances to be normal in , its normal
closure in the absolute elementary subgroup is denoted by
. The main results of [12] implied that the commutator
is in fact normal in . In the present
paper we prove an unexpected result that in fact
. It follows that
the standard commutator formula also holds in the unrelativised form, namely
, where
is the full congruence subgroup of level . In particular,
is normal in .Comment: 14 Page
Grothendieck-Serre Conjecture I: Appendix
We prove here some supplementary statements that appeared without proof in I.
Panin, A. Stavrova, N. Vavilov, On Grothendieck--Serre's conjecture concerning
principal -bundles over reductive group schemes:I, arXiv:0905.1418Comment: We prove some supplementary statements that appeared without proof in
arXiv:0905.1418; 10 page
Generation of relative commutator subgroups in Chevalley groups
Let be a reduced irreducible root system of rank , let be a
commutative ring and let be two ideals of . In the present paper we
describe generators of the commutator groups of relative elementary subgroups
both as normal subgroups of the elementary
Chevalley group , and as groups. Namely, let x_{\a}(\xi),
\a\in\Phi, , be an elementary generator of . As a normal
subgroup of the absolute elementary group , the relative elementary
subgroup is generated by x_{\a}(\xi), \a\in\Phi, . Classical
results due to Michael Stein, Jacques Tits and Leonid Vaserstein assert that as
a group is generated by z_{\a}(\xi,\eta), where \a\in\Phi,
, . In the present paper, we prove the following
birelative analogues of these results. As a normal subgroup of the
relative commutator subgroup is generated
by the following three types of generators: i)
, ii)
, and iii)
, where , , ,
. As a group, the generators are essentially the same, only that
type iii) should be enlarged to iv) . For classical
groups, these results, with much more computational proofs, were established in
previous papers by the authors. There is already an amazing application of
these results, namely in the recent work of Alexei Stepanov on relative
commutator width
Multiple Commutator Formulas for Unitary Groups
Let (\FormR) be a form ring such that is quasi-finite -algebra
(i.e., a direct limit of module finite algebras) with identity. We consider the
hyperbolic Bak's unitary groups \GU(2n,\FormR), . For a form ideal
of the form ring (\FormR) we denote by \EU(2n,I,\Gamma) and
\GU(2n,I,\Gamma) the relative elementary group and the principal congruence
subgroup of level , respectively. Now, let ,
, be form ideals of the form ring . The main result of
the present paper is the following multiple commutator formula
[\big[\EU(2n,I_0,\Gamma_0),&\GU(2n,I_1,\Gamma_1),\GU(2n, I_2,\Gamma_2),...,
\GU(2n,I_m,\Gamma_m)\big]=
&\big[\EU(2n,I_0,\Gamma_0),\EU(2n,I_1,\Gamma_1),\EU(2n,I_2,\Gamma_2),...,
\EU(2n, I_m, \Gamma_m)\big],] which is a broad generalization of the standard
commutator formulas. This result contains all previous results on commutator
formulas for classical like-groups over commutative and finite-dimensional
rings.Comment: arXiv admin note: text overlap with arXiv:0911.551
Relative commutator calculus in Chevalley groups
We revisit localisation and patching method in the setting of Chevalley
groups. Introducing certain subgroups of relative elementary Chevalley groups,
we develop relative versions of the conjugation calculus and the commutator
calculus in Chevalley groups , \rk(\Phi)\geq 2, which are both
more general, and substantially easier than the ones available in the
literature. For classical groups such relative commutator calculus has been
recently developed by the authors in \cite{RZ,RNZ}. As an application we prove
the mixed commutator formula, \big [E(\Phi,R,\ma),C(\Phi,R,\mb)\big ]=\big
[E(\Phi,R,\ma),E(\Phi,R,\mb)\big], for two ideals \ma,\mb\unlhd R. This
answers a problem posed in a paper by Alexei Stepanov and the second author
Relative centralisers of relative subgroups
Let be an associative ring with 1, be the general linear
group of degree over . In this paper we calculate the relative
centralisers of the relative elementary subgroups or the principal congruence
subgroups, corresponding to an ideal modulo the relative elementary
subgroups or the principal congruence subgroups, corresponding to another ideal
. Modulo congruence subgroups the results are essentially easy
exercises in linear algebra. But modulo the elementary subgroups they turned
out to be quite tricky, and we could get definitive answers only over
commutative rings, or, in some cases, only over Dedekind rings. We discuss also
some further related problems, such as the interrelations of various birelative
commutator subgroups, etc., and state several unsolved questions.Comment: 12 page
Commutators of relative and unrelative elementary subgroups in Chevalley groups
In the present paper, which is a direct sequel of our papers [10,11,35] joint
with Roozbeh Hazrat, we achieve a further dramatic reduction of the generating
sets for commutators of relative elementary subgroups in Chevalley groups.
Namely, let be a reduced irreducible root system of rank , let
be a commutative ring and let be two ideals of . We consider
subgroups of the Chevalley group of type over . The
unrelative elementary subgroup of level is generated (as a
group) by the elementary unipotents , , ,
of level . Its normal closure in the absolute elementary subgroup
is denoted by and is called the relative elementary
subgroup of level . The main results of [11,35] consisted in construction of
economic generator sets for the mutual commutator subgroups
, where and are two ideals of . It turned
out that one can take Stein---Tits---Vaserstein generators of ,
plus elementary commutators of the form
, where , .
Here we improve these results even further, by showing that in fact it suffices
to engage only elementary commutators corresponding to {\it one\/} long root,
and that modulo the commutators behave as
symbols. We discuss also some further variations and applications of these
results.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1811.1126
Relative and unrelative elementary groups, revisited
Let be any associative ring with , , and let be
two-sided ideals of . In the present paper we show that the mixed commutator
subgroup is generated as a group by the elements of the
two following forms: 1) and , 2)
, where , , , . Moreover, for the second type of generators, it suffices to fix one pair of
indices . This result is both stronger and more general than the
previous results by Roozbeh Hazrat and the authors. In particular, it implies
that for all associative rings one has the equality
and many further
corollaries can be derived for rings subject to commutativity conditions.Comment: 12 page
Multiple commutators of elementary subgroups: end of the line
In our previous joint papers with Roozbeh Hazrat and Alexei Stepanov we
established commutator formulas for relative elementary subgroups in ,
, and other similar groups, such as Bak's unitary groups, or Chevalley
groups. In particular, there it was shown that multiple commutators of
elementary subgroups can be reduced to double such commutators. However, since
the proofs of these results depended on the standard commutator formulas, it
was assumed that the ground ring is quasi-finite. Here we propose a
different approach which allows to lift any such assumptions and establish
almost definitive results. In particular, we prove multiple commutator
formulas, and other related facts for over an {\it arbitrary}
associative ring .Comment: 14 pages. arXiv admin note: text overlap with arXiv:1910.0898
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