pith. sign in

arxiv: 2601.00379 · v3 · pith:GQ47ABZEnew · submitted 2026-01-01 · 🧮 math.RT

Complete invariants for simultaneous similarity

Klaus Bongartz , Shmuel Friedland This is my paper

Pith reviewed 2026-05-22 11:38 UTC · model grok-4.3

classification 🧮 math.RT
keywords simultaneous similaritymatrix tuplesorbit classificationinvariant morphismsGL_n actionlocally closed decompositionmodule varietiesrepresentation theory
0
0 comments X

The pith

Discrete and continuous invariants completely determine the orbits of p-tuples of n by n matrices under simultaneous similarity over any field.

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

The paper constructs invariants that classify when two collections of square matrices are simultaneously similar under a common change of basis. Discrete invariants first partition the entire space of such collections into finitely many locally closed subsets that remain stable under the group action. On each subset a finite collection of polynomial maps to the base field then separates the orbits inside it. The original action thereby reduces to the simpler left-multiplication action of a product of smaller general linear groups on a product of rectangular matrix spaces. A reader cares because the question of simultaneous similarity appears throughout linear algebra, control theory, and the representation theory of algebras.

Core claim

For the variety of p-tuples of n by n matrices over an arbitrary field k under the simultaneous similarity action of GL_n, discrete invariants decompose the variety into finitely many locally closed GL_n-stable subsets. On each such subset, finitely many invariant morphisms to k separate the orbits. This reduces the complicated similarity action to left multiplication by a product of GL_{l_i}'s on a product of rectangular spaces k^{l_i x m_i}. An analogous complete classification holds for the left-right action of GL_m times GL_n on p-tuples of m by n matrices and, more generally, for the variety of finite-dimensional modules over any finitely generated algebra.

What carries the argument

Discrete invariants that induce a finite decomposition into locally closed GL_n-stable subsets, followed by the explicit reduction of the simultaneous similarity action to products of left multiplications on rectangular matrix spaces.

Load-bearing premise

Discrete invariants can be constructed that produce a finite decomposition of the space of matrix tuples into locally closed subsets stable under the simultaneous similarity action, and this construction works for arbitrary fields, arbitrary p, and arbitrary n.

What would settle it

A concrete counter-example would be a specific field k, numbers p and n, and two matrix tuples lying in distinct orbits that nevertheless receive identical discrete invariants and identical values for all the continuous invariant morphisms in the corresponding subset.

read the original abstract

Always dealing with an arbitrary field we consider the variety $(k^{n\times n})^{p}$ under the action of $GL_{n}$ by simultaneous similarity. We define discrete and continuous invariants which completely determine the orbits. The discrete invariants induce a disjoint decomposition of the variety into finitely many locally closed $GL_{n}$-stable subsets and for each of these we construct finitely many invariant morphisms to $k$ separating the orbits. The complicated action of $GL_{n}$ by similarity is reduced to left multiplication of a product of $GL_{l_{i}}$'s on a product of $k^{l_{i}\times m_{i}}$'s. An analogous result holds for the left-right action of $GL_{m}\times GL_{n}$ on $(k^{m\times n })^{p}$ and more generally for all varieties of finite dimensional modules over some finitely generated algebra.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs complete invariants for the simultaneous similarity action of GL_n(k) on the variety (k^{n×n})^p over an arbitrary field k. Discrete invariants induce a finite disjoint decomposition of the variety into locally closed GL_n-stable subsets; on each subset, finitely many invariant morphisms to k are constructed that separate the orbits. The simultaneous similarity action is thereby reduced to left multiplication by a product of smaller GL_{l_i} groups on a product of rectangular matrices k^{l_i × m_i}. Analogous results are stated for the left-right action of GL_m × GL_n on (k^{m×n})^p and, more generally, for finite-dimensional modules over finitely generated k-algebras.

Significance. If the constructions hold, the work supplies an explicit, uniform method for producing complete invariants for simultaneous similarity over arbitrary fields, reducing a non-linear group action to a collection of simpler linear actions. The extension to modules over arbitrary finitely generated algebras broadens the scope beyond matrix tuples. The paper provides concrete constructions rather than pure existence statements, which is a strength for applications in representation theory and computational algebra.

major comments (1)
  1. The load-bearing step is the uniform construction, for arbitrary k (including finite fields and positive characteristic), of discrete invariants whose level sets are locally closed and finite in number. If this construction is only sketched or relies on properties that fail when k is not algebraically closed, the reduction to rectangular-matrix actions cannot be applied uniformly and the completeness claim does not follow. The manuscript should make the explicit form of these discrete invariants and the proof of local closedness and finiteness fully visible in the main text.
minor comments (2)
  1. Notation for the rectangular matrices k^{l_i × m_i} and the indices l_i, m_i should be introduced with a clear reference to the preceding decomposition (e.g., in the paragraph immediately after the statement of the main theorem).
  2. The general statement for modules over finitely generated algebras is announced but not given a numbered theorem; adding one would clarify the precise scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the central role of the discrete invariants. We address the major comment below and will revise the manuscript to increase the visibility and explicitness of the relevant constructions and proofs.

read point-by-point responses

  1. Referee: The load-bearing step is the uniform construction, for arbitrary k (including finite fields and positive characteristic), of discrete invariants whose level sets are locally closed and finite in number. If this construction is only sketched or relies on properties that fail when k is not algebraically closed, the reduction to rectangular-matrix actions cannot be applied uniformly and the completeness claim does not follow. The manuscript should make the explicit form of these discrete invariants and the proof of local closedness and finiteness fully visible in the main text.

    Authors: The discrete invariants are constructed explicitly in Section 3 as the tuple of ranks of all matrices obtained by evaluating non-commutative polynomials of bounded degree on the given matrix tuple; these ranks are given by the vanishing of certain minors and are therefore polynomial functions defined over any field k. The level sets are the intersections of the sets where a given collection of minors vanish and the complementary minors do not vanish; each such set is therefore locally closed in the Zariski topology over arbitrary k. Finiteness follows immediately from the fact that each rank is an integer between 0 and n. The subsequent reduction to rectangular-matrix actions is carried out on each level set by choosing bases adapted to the flag defined by the kernels and images of these polynomials; this choice is possible over any field once the ranks are fixed. We agree that the current exposition places some of the verification in the proofs of the main theorems rather than in a dedicated preliminary subsection. In the revised version we will add an expanded subsection (new Section 2.2) that isolates the construction, states the local-closedness and finiteness claims as separate lemmas, and proves them using only elementary linear algebra over arbitrary fields, without any appeal to algebraic closure. revision: yes

Circularity Check

0 steps flagged

Explicit construction of discrete invariants and reduction to simpler actions with no circularity

full rationale

The paper's derivation starts from an explicit definition of discrete invariants on (k^{n×n})^p under simultaneous similarity, which by construction induce a finite disjoint decomposition into locally closed GL_n-stable subsets for arbitrary fields. Each piece then reduces directly to left multiplication actions of products of smaller GL_{l_i} on rectangular matrices k^{l_i × m_i}, after which finitely many invariant morphisms to k are constructed to separate orbits. This chain is self-contained: the invariants and decomposition are not defined in terms of the orbits they classify, no parameters are fitted to data and renamed as predictions, and no load-bearing step relies on self-citation or imported uniqueness theorems. The abstract and described construction present a direct algorithmic reduction rather than a tautology or renaming of known patterns. The result is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard facts from algebraic geometry and representation theory about group actions on varieties and the existence of separating invariants on locally closed subsets. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math The variety (k^{n×n})^p carries a natural algebraic action of GL_n by simultaneous similarity.
    Invoked in the first sentence of the abstract as the setting for the orbit classification problem.
  • domain assumption Algebraic group actions on affine varieties admit finite decompositions into locally closed stable subsets on which separating invariants exist.
    This background fact from geometric invariant theory is presupposed to guarantee the finite decomposition and the construction of invariant morphisms to k.

pith-pipeline@v0.9.0 · 5674 in / 1493 out tokens · 36105 ms · 2026-05-22T11:38:56.019641+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Degeneration order of $3\times 3$ nilpotent matrix tuples

    math.RT 2026-04 unverdicted novelty 6.0

    The degeneration order of simultaneous similarity classes of 3×3 nilpotent matrix tuples is given by rank conditions.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [1]

    Arnol’d: Arnold’s problems

    V.I. Arnol’d: Arnold’s problems. Second edition 2003, Springer

    work page 2003

  2. [2]

    Assem; D

    I. Assem; D. Simson; A. Skowro´ nski: Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory. London Mathematical Society Student Texts 65

    work page

  3. [3]

    Baur: Decidability and undecidability of theories of Abelian groups with predicates for subgroups

    W. Baur: Decidability and undecidability of theories of Abelian groups with predicates for subgroups. Compos. Math. 31, 23-30 (1975)

    work page 1975

  4. [4]

    Bongartz: Schichten von Matrizen sind rationale Variet¨ aten, Math.Ann.283,53-64 (1989)

    K. Bongartz: Schichten von Matrizen sind rationale Variet¨ aten, Math.Ann.283,53-64 (1989)

    work page 1989

  5. [5]

    Bongartz: Gauß-Elimination und der gr¨ oßte gemeinsame direkte Summand von zwei endlichdimensionalen Moduln, Arch.Math.,Vol

    K. Bongartz: Gauß-Elimination und der gr¨ oßte gemeinsame direkte Summand von zwei endlichdimensionalen Moduln, Arch.Math.,Vol. 53, 256-258 (1989 )

    work page 1989

  6. [6]

    Bongartz: A remark on Friedlands stratification of varieties of modules, Communications in algebra,23(6),2163-2165 ( 1995 )

    K. Bongartz: A remark on Friedlands stratification of varieties of modules, Communications in algebra,23(6),2163-2165 ( 1995 )

    work page 1995

  7. [7]

    Bongartz: Some geometric aspects of representation theory

    K. Bongartz: Some geometric aspects of representation theory. Algebras and modules I. CMS Conf. Proc. 23, 1-27 (1998)

    work page 1998

  8. [8]

    Bongartz;D

    K. Bongartz;D. Dudek: Decomposition classes of representations of tame quivers. J. Algebra 240, No. 1, 268-288 (2001)

    work page 2001

  9. [9]

    Bongartz: Representation embeddings and the second Brauer-Thrall conjecture

    K. Bongartz: Representation embeddings and the second Brauer-Thrall conjecture. Preprint 32 pages, arXiv:1611.02017 [math.RT] (2016)

    work page arXiv 2016

  10. [10]

    Bongartz:On normal forms for the similarity classes of matrices and pairs of matrices

    K. Bongartz:On normal forms for the similarity classes of matrices and pairs of matrices. 10 pages, arXiv:2502.10926 (2025)

    work page arXiv 2025

  11. [11]

    Brenner: Decomposition properties of some small diagrams of modules

    S. Brenner: Decomposition properties of some small diagrams of modules. Symp. math. 13, Gruppi abeliani, Gruppi e loro rappresent., Convegni 1972, 127-141 (1974)

    work page 1972

  12. [12]

    Borho; H

    W. Borho; H. Kraft: ¨Uber Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helv. 54, 61-104 (1979)

    work page 1979

  13. [13]

    Brooksbank; E.M

    P.A. Brooksbank; E.M. Luks: Testing isomorphism of modules. J. Algebra 320, No. 11, 4020-4029 (2008)

    work page 2008

  14. [14]

    Derksen; I

    H. Derksen; I. Klep; V. Makam; J.Volvciˇ c: Ranks of linear matrix pencils separate simultaneous similarity orbits. Adv. Math. 415, 20 p. (2023)

    work page 2023

  15. [15]

    Formanek: The center of the ring of 4×4 generic matrices

    E. Formanek: The center of the ring of 4×4 generic matrices. J. Algebra 62, 304-319 (1980)

    work page 1980

  16. [16]

    Friedland: Classification of linear systems

    S. Friedland: Classification of linear systems. Proc. of A.M.S. Conf. on Linear Algebra and Its Role in Systems Theory. Contemp. Math. 47, 131-147 (1985)

    work page 1985

  17. [17]

    Friedland: Simultaneous similarity of matrices

    S. Friedland: Simultaneous similarity of matrices. Adv. Math. 50, 189-265 (1983)

    work page 1983

  18. [18]

    Friedland: Simultaneous similarity of matrices

    S. Friedland: Simultaneous similarity of matrices. Bull. Am. Math. Soc., New Ser. 8, 93-95 (1983)

    work page 1983

  19. [19]

    Friedland,Matrices: Algebra, Analysis and Applications, World Scientific, 596 pp., 2016, Singapore

    S. Friedland,Matrices: Algebra, Analysis and Applications, World Scientific, 596 pp., 2016, Singapore

    work page 2016

  20. [20]

    Gelfand; V.A

    I.M. Gelfand; V.A. Ponomarev: Remarks on the classification of a pair of commuting linear transformations in a finite-dimensional space. (Russian) Funkcional. Anal. i Priloˇzen. 3 1969 no. 4, 81-82

    work page 1969

  21. [21]

    Hartshorne: Algebraic Geometry

    R. Hartshorne: Algebraic Geometry. Springer-Verlag 1997

    work page 1997

  22. [22]

    J. E. Humphreys: Linear Algebraic Groups, Graduate texts in mathematics, Springer 1995

    work page 1995

  23. [23]

    Kemper: The computation of invariant fields and a constructive version of a theorem by Rosenlicht

    G. Kemper: The computation of invariant fields and a constructive version of a theorem by Rosenlicht. Transform. Groups 12, No. 4, 657-670 (2007)

    work page 2007

  24. [24]

    H. Kraft;P. Slodowy; T.A. Springer (ed.); Algebraic transformation groups and invariant theory. DMV Seminar, 13. Basel etc.: Birkhr¨ auser Verlag. 211 S. DM 58.00 (1989)

    work page 1989

  25. [25]

    Mumford and J

    D. Mumford and J. Fogarty: Geometric invariant theory,Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer Verlag 1982

    work page 1982

  26. [26]

    Procesi: The invariant theory ofn×nmatrices

    C. Procesi: The invariant theory ofn×nmatrices. Adv. Math. 19, 306-381 (1976)

    work page 1976

  27. [27]

    Rosenlicht: Some basic theorems on algebraic groups

    M. Rosenlicht: Some basic theorems on algebraic groups. Amer. J. Math. 78 401–443 (1956)

    work page 1956

  28. [28]

    Rosenlicht: A remark on quotient spaces

    M. Rosenlicht: A remark on quotient spaces. Anais Acad. Brasil. Ci. 35, 487-489 (1963)

    work page 1963

  29. [29]

    Seshadri: Some results on the quotient space by an algebraic group of automorphisms

    C.S. Seshadri: Some results on the quotient space by an algebraic group of automorphisms. Math. Ann. 149, 286-301 (1963)

    work page 1963

  30. [30]

    Forbregd; N.M

    T.A. Forbregd; N.M. Nornes; S.O. Smalø: Partial orders on representations of algebras. J. Algebra 323, No. 7, 2058-2062 (2010). Universit¨at Wuppertal,Wuppertal, Germany,klausbongartz@online.de Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607-7045, USA,friedlan@uic.edu

    work page 2058