arxiv: 2605.09766 · v1 · submitted 2026-05-10 · 🧮 math.AG

Recognition: 1 theorem link

· Lean Theorem

Centralizers of the complex orthogonal and symplectic group

Tadej Star\v{c}i\v{c}

Pith reviewed 2026-05-12 02:57 UTC · model grok-4.3

classification 🧮 math.AG

keywords centralizersorthogonal groupssymplectic groupsisotropy groupsblock Toeplitz matricessimilarity transformationsskew-symmetric matricesHamiltonian matrices

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The pith

The centralizers of complex orthogonal and symplectic groups admit a recursive block-Toeplitz description under similarity.

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

This paper develops a recursive algorithm to determine the precise centralizers of the complex orthogonal and symplectic groups. These centralizers appear when considering how these groups act by similarity on spaces of skew-symmetric matrices and Hamiltonian matrices. Determining the centralizers also identifies the isotropy groups for these actions. A sympathetic reader would care because such centralizers reveal the symmetry and orbit structure in these matrix spaces under group actions.

Core claim

We find a recursive algorithm for computing the precise centralizers of the complex orthogonal and symplectic groups, and hence the isotropy groups, with respect to the similarity transformation on the spaces of skew-symmetric and Hamiltonian matrices, respectively. These groups are conjugate to groups of certain nonsingular block matrices whose blocks are rectangular block Toeplitz.

What carries the argument

Conjugation of the centralizer groups to groups of nonsingular block matrices whose blocks are rectangular block Toeplitz, which supports the recursive computation.

If this is right

The isotropy groups for the similarity actions on skew-symmetric and Hamiltonian matrices can be computed recursively as well.
Centralizer computations become systematic through reduction to the block matrix structure.
The description supplies exact algorithmic access to the groups rather than only their dimensions.

Where Pith is reading between the lines

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

The recursive structure could be implemented computationally to calculate centralizers for matrices of moderate size.
The same conjugation technique might be tested on other classical groups to check for analogous block forms.

Load-bearing premise

The centralizers admit a precise recursive description via conjugation to nonsingular block matrices with rectangular block Toeplitz blocks.

What would settle it

An explicit matrix belonging to one of the centralizers whose form after conjugation fails to consist of nonsingular blocks with rectangular block Toeplitz structure would disprove the recursive description.

read the original abstract

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

The abstract claims a recursive algorithm for centralizers of the complex orthogonal and symplectic groups via block-Toeplitz conjugation but supplies no derivation, examples, or verification.

read the letter

The main takeaway is that this paper announces a recursive algorithm to compute the centralizers of the complex orthogonal and symplectic groups under similarity on skew-symmetric and Hamiltonian matrices, along with the claim that these centralizers are conjugate to groups of nonsingular block matrices with rectangular block Toeplitz blocks, yielding the isotropy groups as well. If the construction holds, it could give a concrete computational tool for explicit calculations in this corner of algebraic geometry and representation theory. That kind of practical method is what the abstract highlights as new, and it is not presented as a restatement of earlier work. The paper does well to focus on an algorithmic description rather than pure existence, which aligns with needs for hands-on computations. The central soft spot is the complete absence of supporting material. There are no equations, no base cases for the recursion, no low-dimensional checks, and no outline of how the conjugation to the block-Toeplitz form is derived or why the blocks must be rectangular. Without any of that, it is impossible to tell whether the claimed structure is accurate or if the recursion is well-defined and effective. The assumption that such a precise recursive description exists is stated as fact but left unexamined in the available text. This work would mainly interest specialists who already compute centralizers or isotropy groups for these actions and want a new procedure to try. A reader in representation theory or matrix group calculations might extract value if the full paper delivers the steps and some validation. I would recommend sending the full version to peer review, because a working recursive algorithm here could be useful even if it requires heavy checking and revision. As it stands with only the abstract, there is not enough to assess the math or the novelty claim.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to provide a recursive algorithm for computing the precise centralizers of the complex orthogonal and symplectic groups (and hence the isotropy groups) under similarity transformations acting on the spaces of skew-symmetric and Hamiltonian matrices, respectively. It further asserts that these centralizers are conjugate to groups of nonsingular block matrices whose blocks are rectangular block Toeplitz.

Significance. If the claimed recursive description and conjugacy hold, the result would supply a concrete computational tool for determining centralizers and stabilizers in these classical settings, potentially simplifying orbit computations in algebraic geometry and Lie theory. The block-Toeplitz structure, if verified, could connect to existing work on structured matrices and inductive descriptions of centralizers.

major comments (1)

[Abstract] Abstract: The central claim—that a recursive algorithm exists via conjugation to nonsingular block matrices with rectangular block Toeplitz blocks—is asserted without any derivation, base cases, inductive step, explicit matrices, low-dimensional verification, or group-action computation. This absence makes it impossible to assess whether the algorithm is correct or whether the stated conjugacy holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and comments on our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim—that a recursive algorithm exists via conjugation to nonsingular block matrices with rectangular block Toeplitz blocks—is asserted without any derivation, base cases, inductive step, explicit matrices, low-dimensional verification, or group-action computation. This absence makes it impossible to assess whether the algorithm is correct or whether the stated conjugacy holds.

Authors: We agree that the provided abstract summarizes the main result without including derivations, base cases, inductive steps, explicit matrices, low-dimensional verifications, or group-action computations. The current manuscript text consists solely of this abstract. To allow proper assessment of the claims, we will revise the manuscript by expanding it to include a detailed description of the recursive algorithm, along with the necessary base cases, inductive steps, explicit matrix constructions, low-dimensional examples, and group-action computations. revision: yes

Circularity Check

0 steps flagged

No circularity: abstract states algorithmic result without equations or self-referential steps

full rationale

The abstract claims a recursive algorithm for centralizers and a conjugation to block-Toeplitz matrices but supplies no equations, inductive steps, base cases, or derivation chain. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. The result is presented as a construction to be derived elsewhere; nothing reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts from algebraic group theory and linear algebra concerning centralizers and similarity actions; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math Standard properties of complex orthogonal and symplectic groups under similarity transformations hold and admit recursive centralizer computation.
Invoked implicitly to justify the existence of the algorithm and the block Toeplitz conjugacy.

pith-pipeline@v0.9.0 · 5304 in / 1226 out tokens · 50493 ms · 2026-05-12T02:57:26.418031+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

De Teran, F

F. De Teran, F. M. Dopico, The solution of the equationXA+AXT = 0 and its application to the theory of orbits, Lin. Alg. Appl. 434 (1), 44-67

work page
[2]

Dmytryshyn, B

A. Dmytryshyn, B. K ˚agstr¨om, Coupled Sylvester-type Matrix Equations and Block Di- agonalization, SIAM J. Matrix Anal. Appl. 36 (No. 2) (2015), 580-593

work page 2015
[3]

Dmytryshyn, B

A. Dmytryshyn, B. K ˚agstr¨om, V. V. Sergeichuk, Skew-symmetric matrix pencils: Codi- mension counts and the solution of a pair of matrix equations, Linear Algebra Appl. 438 (No. 8) (2013), 3375-3396

work page 2013
[4]

D. ˇZ. ¯Dokovi´v, K. Rietsch, K. Zhao, Normal forms for orthogonal similarity classes of skew-symmetric matrices, J. Algebra 308 (2007) 686-703

work page 2007
[5]

F. R. Gantmacher, The theory of matrices, Chelsea Publishing Company, New York, 1959

work page 1959
[6]

R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990

work page 1990
[7]

R. A. Horn, D. I. Merino, Contragradient equivalence: a canonical form and some ap- plications, Linear Algebra Appl., 214 (1995), pp. 43-92

work page 1995
[8]

L. K. Hua, Geometries of matrices I. Generalizations of von Staudt’s theorem, Trans. Amer. Math. Soc. 57 (1945), 441-481

work page 1945
[9]

Lancaster, L

P . Lancaster, L. Rodman, Algebraic Riccati Equations, Clarendon Press, Oxford, 1995

work page 1995
[10]

A. J. Laub, K. Meyer, Canonical forms for symplectic and Hamiltonian matrices. Ce- lestial Mechanics 9 (1974), 213-238

work page 1974
[11]

W. W. Lin, V. Mehrmann, H. Xu, Canonical Forms for Hamiltonian and Symplectic Matrices and Pencils, Linear Algebra Appl. 302-303 (1999), 469-533

work page 1999
[12]

W., Liebeck, G

M. W., Liebeck, G. M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and lie algebras, Providence, American Mathematical Society, 2012. 20 TADEJ STAR ˇCI ˇC

work page 2012
[13]

Mehl, On classification of normal matrices in indefinite inner product spaces, Elec- tron

C. Mehl, On classification of normal matrices in indefinite inner product spaces, Elec- tron. J. Linear Algebra 15 (2006), 50-83

work page 2006
[14]

J. S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge: Cambridge University Press, 2017

work page 2017
[15]

Springer, R

T.A. Springer, R. Steinberg, Conjugacy classes. In: Seminar on Algebraic Groups and Related Finite Groups. Lecture Notes in Mathematics, vol 131. Springer, Berlin, Hei- delberg, 1970

work page 1970
[16]

Star ˇciˇc, Hong’s canonical form of a Hermitian matrix with respect to orthogonal *congruence, Linear Algebra Appl

T. Star ˇciˇc, Hong’s canonical form of a Hermitian matrix with respect to orthogonal *congruence, Linear Algebra Appl. 630 (2021), 241-251

work page 2021
[17]

Star ˇciˇc, Isotropy groups of the action of orthogonal *congruence on Hermitian ma- trices, Linear Algebra Appl

T. Star ˇciˇc, Isotropy groups of the action of orthogonal *congruence on Hermitian ma- trices, Linear Algebra Appl. 694 (2024), 101-135

work page 2024
[18]

Star ˇciˇc, Isotropy groups of the action of orthogonal similarity on symmetric matri- ces, Linear Multilinear Algebra

T. Star ˇciˇc, Isotropy groups of the action of orthogonal similarity on symmetric matri- ces, Linear Multilinear Algebra. 71 (no. 5) (2023), 842-866

work page 2023
[19]

Wan, Geometry of matrices, World Scientific, New York Heidelberg Berlin, 1996

Z. Wan, Geometry of matrices, World Scientific, New York Heidelberg Berlin, 1996

work page 1996
[20]

Wellstein, ¨Uber symmetrische, alternierende und orthogonale Normalformen von Matrizen

J. Wellstein, ¨Uber symmetrische, alternierende und orthogonale Normalformen von Matrizen. J. f¨ur Reine Angew. Math., vol. 1930, no. 163, 1930, pp. 166-182

work page 1930
[21]

Weyl, The classical groups, their invariants and representations, Princeton Univer- sity Press, 1946

H. Weyl, The classical groups, their invariants and representations, Princeton Univer- sity Press, 1946. Faculty of Education, University of Ljubljana, Kardeljeva Ploˇsˇcad 16, 1000 Ljubljana, Slove- nia Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia Email address:tadej.starcic@pef.uni-lj.si

work page 1946