Another presentation of orthogonal Steinberg groups

Egor Voronetsky

arxiv: 2012.12147 · v3 · submitted 2020-12-22 · 🧮 math.GR

Another presentation of orthogonal Steinberg groups

Egor Voronetsky This is my paper

Pith reviewed 2026-05-24 14:09 UTC · model grok-4.3

classification 🧮 math.GR

keywords orthogonal Steinberg groupspro-group approachvan der Kallen presentationESD-transvectionsquadratic formscommutative ringsSteinberg groups

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The pith

The orthogonal Steinberg group StO(M, q) admits van der Kallen's another presentation when the quadratic form q is sufficiently isotropic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the pro-group approach to establish that the orthogonal Steinberg group StO(M, q) has van der Kallen's another presentation. This applies when M is a module over a commutative ring and q is a sufficiently isotropic quadratic form on M. The authors also build analogs of ESD-transvections inside the corresponding orthogonal Steinberg pro-groups, given assumptions on the parameters. A sympathetic reader would care because an alternative presentation can make the structure and relations of these groups easier to handle in computations or further proofs.

Core claim

Using the pro-group approach, the paper shows that StO(M, q) admits van der Kallen's another presentation, where M is a module over a commutative ring with sufficiently isotropic quadratic form q. It further constructs an analog of ESD-transvections in orthogonal Steinberg pro-groups under some assumptions on their parameters.

What carries the argument

The pro-group approach applied to orthogonal Steinberg groups, which produces the alternative presentation.

If this is right

StO(M, q) can be worked with using the generators and relations from the another presentation.
Analog ESD-transvections exist and can be used inside the orthogonal Steinberg pro-groups.
The result applies precisely when the isotropy condition on q holds and the parameter assumptions are met.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same pro-group method might produce similar presentations for other classical Steinberg groups beyond the orthogonal case.
The alternative presentation could simplify calculations of K_1 or related functors attached to these groups.
It may be possible to relax the isotropy condition by adjusting the pro-group construction.

Load-bearing premise

The quadratic form q on M must be sufficiently isotropic so that the pro-group construction yields the presentation.

What would settle it

An explicit commutative ring, module M, and quadratic form q that is not sufficiently isotropic, yet StO(M, q) still satisfies van der Kallen's another presentation exactly.

read the original abstract

We use the pro-group approach to show that $\mathrm{StO}(M, q)$ admits van der Kallen's "another presentation", where $M$ is a module over a commutative ring with sufficiently isotropic quadratic form $q$. Moreover, we construct an analog of ESD-transvections in orthogonal Steinberg pro-groups under some assumptions on their parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper uses the pro-group method to extend van der Kallen's presentation to orthogonal Steinberg groups under the isotropy condition.

read the letter

The key point is that Voronetsky shows StO(M, q) has van der Kallen's another presentation by means of pro-groups, when q is sufficiently isotropic on the module M over a commutative ring. He also constructs an analog of ESD-transvections in the orthogonal pro-group setting with some assumptions on the parameters. This is new as an application to the orthogonal case. The pro-group approach had been used elsewhere, and here it yields the desired presentation plus the transvection analog, which fills a gap for this family of groups. The work is done well in that the assumptions are stated explicitly in the abstract, avoiding overstatement. The isotropy condition is the standard one needed for these groups to have good generation properties, so the result aligns with what one would expect. Soft spots are limited. The main one is the dependence on the isotropy and parameter conditions, which restricts the rings and forms where this applies directly. That's proportionate to the claim and not a hidden issue. There is no indication of circularity or fitting data to the result. Since the full proof is not in the abstract, one would want to check the details in review, but the outline is consistent. This paper is for people working on Steinberg groups and presentations in algebraic K-theory. A specialist in that area would get value from the explicit construction and the extension. It is not broad enough for a general audience. I recommend sending it for peer review. The result is a legitimate technical advance with clear boundaries, and the method is reproducible in principle.

Referee Report

0 major / 2 minor

Summary. The manuscript uses the pro-group approach to prove that the orthogonal Steinberg group StO(M, q) admits van der Kallen's 'another presentation' when M is a module over a commutative ring equipped with a sufficiently isotropic quadratic form q. It additionally constructs an analog of ESD-transvections in orthogonal Steinberg pro-groups under assumptions on the parameters.

Significance. If the central claim holds, the result extends the pro-group method from other Steinberg groups to the orthogonal case, supplying an alternative presentation that may facilitate computations or structural results in algebraic K-theory and the theory of groups over rings. The ESD-transvection analog further enriches the toolkit for these pro-groups.

minor comments (2)

[Abstract] Abstract: the phrase 'sufficiently isotropic' is used without a precise definition or forward reference to the section where the condition is stated; this should be clarified for readers.
[Introduction] The manuscript should include a brief comparison, even if only in the introduction, between the pro-group presentation obtained here and the original van der Kallen presentation to highlight the differences.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external pro-group method to known presentation

full rationale

The paper states it uses the pro-group approach to establish that StO(M, q) admits van der Kallen's another presentation when q is sufficiently isotropic. This is a direct proof claim relying on an external method and explicit assumptions on isotropy and ESD-transvection parameters. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or description. The central result does not reduce to its inputs by construction, and the approach is presented as independent of the target presentation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard ring and module theory plus the pro-group framework; no free parameters, invented entities, or ad-hoc axioms are visible.

axioms (1)

domain assumption Commutative rings and modules admit sufficiently isotropic quadratic forms q under which the pro-group construction works.
The central claim is conditioned on this isotropy assumption stated in the abstract.

pith-pipeline@v0.9.0 · 5563 in / 1139 out tokens · 27607 ms · 2026-05-24T14:09:58.523117+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

S. Böge. Steinberggruppen von orthogonalen gruppen. J. Reine Angew. Math., 494:219–236, 1998

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[2]

Lavrenov

A. Lavrenov. Another presentation for symplectic Stein berg groups. J. Pure Appl. Alg. , 219(9):3755–3780, 2015

work page 2015
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Lavrenov and S

A. Lavrenov and S. Sinchuk. On centrality of even orthogo nal K2. J. Pure Appl. Alg. , 221(5):1134–1145, 2017

work page 2017
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Lavrenov, S

A. Lavrenov, S. Sinchuk, and E. Voronetsky. Centrality o f odd unitary K2-functor, 2020. Preprint, arXiv:2009.03999

work page arXiv 2020
[5]

B. A. Magurn, W. Van der Kallen, and L. N. Vaserstein. Abso lute sta- ble rank and Witt cancellation for noncommutative rings. Invent. Math. , 91:525–542, 1988

work page 1988
[6]

V. Petrov. Odd unitary groups. J. Math. Sci. , 130(3):4752–4766, 2005

work page 2005
[7]

S. Sinchuk. On centrality of K2 for Chevalley groups of type El. J. Pure Appl. Alg. , 220(2):857–875, 2016

work page 2016
[8]

Tulenbaev

M.S. Tulenbaev. Schur multiplier of the group of element ary matrices of ﬁnite order. J. Sov. Math. , 17(4):2062–2067, 1981

work page 2062
[9]

van der Kallen

W. van der Kallen. Another presentation for Steinberg gr oups. Indag. Math., 80(4):304–312, 1977

work page 1977
[10]

Voronetsky

E. Voronetsky. Centrality of K2-functor revisited. J. Pure Appl. Alg. , 225(4), 2020. published electronically

work page 2020
[11]

Voronetsky

E. Voronetsky. Centrality of odd unitary K2-functor, 2020. Preprint, arXiv:2005.02926

work page arXiv 2020
[12]

Voronetsky

E. Voronetsky. Injective stability for odd unitary K1. J. Group Theory , 23(5), 2020

work page 2020
[13]

M. Wendt. On homotopy invariance for homology of rank tw o groups. J. Pure Appl. Alg. , 216(10):2291–2301, 2012. 11

work page 2012

[1] [1]

S. Böge. Steinberggruppen von orthogonalen gruppen. J. Reine Angew. Math., 494:219–236, 1998

work page 1998

[2] [2]

Lavrenov

A. Lavrenov. Another presentation for symplectic Stein berg groups. J. Pure Appl. Alg. , 219(9):3755–3780, 2015

work page 2015

[3] [3]

Lavrenov and S

A. Lavrenov and S. Sinchuk. On centrality of even orthogo nal K2. J. Pure Appl. Alg. , 221(5):1134–1145, 2017

work page 2017

[4] [4]

Lavrenov, S

A. Lavrenov, S. Sinchuk, and E. Voronetsky. Centrality o f odd unitary K2-functor, 2020. Preprint, arXiv:2009.03999

work page arXiv 2020

[5] [5]

B. A. Magurn, W. Van der Kallen, and L. N. Vaserstein. Abso lute sta- ble rank and Witt cancellation for noncommutative rings. Invent. Math. , 91:525–542, 1988

work page 1988

[6] [6]

V. Petrov. Odd unitary groups. J. Math. Sci. , 130(3):4752–4766, 2005

work page 2005

[7] [7]

S. Sinchuk. On centrality of K2 for Chevalley groups of type El. J. Pure Appl. Alg. , 220(2):857–875, 2016

work page 2016

[8] [8]

Tulenbaev

M.S. Tulenbaev. Schur multiplier of the group of element ary matrices of ﬁnite order. J. Sov. Math. , 17(4):2062–2067, 1981

work page 2062

[9] [9]

van der Kallen

W. van der Kallen. Another presentation for Steinberg gr oups. Indag. Math., 80(4):304–312, 1977

work page 1977

[10] [10]

Voronetsky

E. Voronetsky. Centrality of K2-functor revisited. J. Pure Appl. Alg. , 225(4), 2020. published electronically

work page 2020

[11] [11]

Voronetsky

E. Voronetsky. Centrality of odd unitary K2-functor, 2020. Preprint, arXiv:2005.02926

work page arXiv 2020

[12] [12]

Voronetsky

E. Voronetsky. Injective stability for odd unitary K1. J. Group Theory , 23(5), 2020

work page 2020

[13] [13]

M. Wendt. On homotopy invariance for homology of rank tw o groups. J. Pure Appl. Alg. , 216(10):2291–2301, 2012. 11

work page 2012