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arxiv: 2606.02805 · v1 · pith:2MZNRI3Znew · submitted 2026-06-01 · 🧮 math.OC · cs.DS· math.AG· math.RT

On the gap of quiver representations

John Maar This is my paper

Pith reviewed 2026-06-28 13:13 UTC · model grok-4.3

classification 🧮 math.OC cs.DSmath.AGmath.RT
keywords quiver representationsnullcone membershipgap condition numberorbit closuresemistabilityinvariant theorycomputational complexity
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The pith

The inverse gap is polynomially bounded in the number of vertices and maximum dimension for type A, Â and uniform tree quivers.

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

The paper establishes that for representations of type A and  quivers, and for tree quivers with uniform dimension vectors, the inverse gap remains bounded by a polynomial in the number of vertices and the largest dimension appearing in the vector. This bound controls the running time of geodesic optimization methods for deciding whether the origin lies in the closure of an orbit under the special linear group. The same polynomial control extends to the decision problem for σ-semistability. For arbitrary quivers the gap can drop exponentially with the number of leaves, and the weight margin can drop exponentially with the number of vertices in any connected quiver.

Core claim

We prove that the inverse gap is polynomially bounded in the number of vertices and the maximum dimension for type A and Â, as well as tree quivers with uniform dimension vectors. Consequently the geodesic optimization algorithm solves the nullcone membership problem in polynomial time for these families. In contrast we construct families where the gap is exponentially small in the number of leaves, and for every connected quiver we exhibit dimension vectors making the weight margin exponentially small in the number of vertices. The results extend to σ-semistability.

What carries the argument

The gap, a condition number of the representation that governs the convergence rate of geodesic optimization for the nullcone membership problem.

If this is right

  • Nullcone membership for the listed quiver families is decidable in time polynomial in the input size.
  • The same polynomial-time guarantee holds for deciding σ-semistability on those families.
  • The gap can be exponentially small in the number of leaves for some quiver families.
  • The weight margin is exponentially small in the number of vertices for dimension vectors on any connected quiver.

Where Pith is reading between the lines

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

  • Similar polynomial bounds might hold for other Dynkin diagrams or for quivers of bounded treewidth.
  • Any algorithm that works for all quivers must tolerate exponentially small condition numbers on some inputs.
  • The polynomial bound separates the computational complexity of the listed families from the general case.

Load-bearing premise

The gap is defined and satisfies the same quantitative properties used by geodesic optimization methods for self-adjoint group actions.

What would settle it

An explicit type A quiver whose number of vertices is n and whose dimensions are at most d, yet whose gap is smaller than any fixed polynomial in n and d.

read the original abstract

The nullcone membership problem, deciding whether an orbit closure contains the origin, is fundamental in computational invariant theory. For self-adjoint groups, B\"urgisser, Franks, Garg, Oliveira, Walter and Wigderson gave a geodesic optimization algorithm whose complexity is controlled by the gap, a condition number of the representation. We study the gap for quiver representations under the action of the special linear group. We prove that the inverse gap is polynomially bounded in the number of vertices and the maximum dimension for type A and $\hat{A}$, as well as tree quivers with uniform dimension vectors. Consequently, the algorithm of B\"urgisser et al. solves the nullcone membership problem in polynomial time for these families. In contrast, we construct families of quivers and dimension vectors where the gap is exponentially small in the number of leaves, furthermore, for every connected quiver we exhibit dimension vectors such that the weight margin (a related condition number) is exponentially small in the number of vertices. We also extend our results to $\sigma$-semistability, thereby giving a new proof of a recent result of Iwamasa, Oki, and Soma.

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

0 major / 2 minor

Summary. The paper analyzes the gap (a condition number) for quiver representations under the SL action in the context of the nullcone membership problem. It proves that the inverse gap is polynomially bounded in the number of vertices and maximum dimension for type A and  quivers as well as tree quivers with uniform dimension vectors; this yields polynomial-time solvability of nullcone membership via the geodesic optimization algorithm of Bürgisser et al. The manuscript also constructs families with exponentially small gaps (in the number of leaves) and shows that the weight margin is exponentially small in the number of vertices for every connected quiver. Finally, it extends the results to σ-semistability, providing a new proof of a theorem of Iwamasa, Oki, and Soma.

Significance. If the stated polynomial bounds hold, the work supplies the first polynomial-time algorithms for nullcone membership on these quiver families, which is a concrete advance in computational invariant theory. The explicit exponential-gap constructions and the weight-margin lower bounds delineate the boundary between tractable and intractable cases. The extension to σ-semistability supplies an independent proof of an existing result and broadens the applicability of the gap analysis.

minor comments (2)
  1. [Abstract] The abstract states that the inverse gap is 'polynomially bounded' without indicating the degree; adding the explicit degree (if obtained in the proofs) would make the complexity claim sharper.
  2. [§1] Notation for the gap and weight margin should be cross-referenced explicitly to the definitions in Bürgisser et al. at the first appearance in §1 to aid readers unfamiliar with the geodesic-optimization framework.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, detailed summary of our contributions, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper imports the gap definition and algorithm from Bürgisser et al. (distinct authors) as a fixed external condition number, then supplies independent mathematical proofs establishing polynomial bounds on the inverse gap for type A, Â, and uniform tree quivers. These bounds rest on standard quiver representation facts over algebraically closed fields of char 0 and explicit constructions, without any reduction of the claimed results to fitted parameters, self-definitions, or load-bearing self-citations. The extension to σ-semistability is likewise a direct re-proof of an external result. No step equates a prediction to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters, invented entities, or ad-hoc axioms beyond standard domain assumptions from representation theory and the cited optimization framework.

axioms (1)
  • domain assumption The gap is defined exactly as in the geodesic optimization algorithm of Bürgisser et al.
    All complexity claims rest on this prior definition of the gap as a condition number.

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Reference graph

Works this paper leans on

29 extracted references · 12 canonical work pages

  1. [1]

    Peter Bürgisser and Cole Franks and Ankit Garg and Rafael Oliveira and Michael Walter and Avi Wigderson , title =. 2019. doi:10.1109/focs.2019.00055 , url =

    work page doi:10.1109/focs.2019.00055 2019

  2. [2]

    36th Computational Complexity Conference,

    William Cole Franks and Philipp Reichenbach , title =. 36th Computational Complexity Conference,. 2021 , url =. doi:10.4230/LIPICS.CCC.2021.13 , timestamp =

    work page doi:10.4230/lipics.ccc.2021.13 2021

  3. [3]

    2017 , publisher=

    An Introduction to Quiver Representations , author=. 2017 , publisher=

    2017

  4. [4]

    Ranks of linear matrix pencils separate simultaneous similarity orbits , JOURNAL =

    Derksen, Harm and Klep, Igor and Makam, Visu and Vol. Ranks of linear matrix pencils separate simultaneous similarity orbits , JOURNAL =. 2023 , PAGES =. doi:10.1016/j.aim.2023.108888 , URL =

    work page doi:10.1016/j.aim.2023.108888 2023

  5. [5]

    Algebra Number Theory , FJOURNAL =

    Derksen, Harm and Makam, Visu , TITLE =. Algebra Number Theory , FJOURNAL =. 2020 , NUMBER =. doi:10.2140/ant.2020.14.2791 , URL =

    work page doi:10.2140/ant.2020.14.2791 2020

  6. [6]

    and Knyazev, A.V

    Zhu, P. and Knyazev, A.V. , year=. Angles between subspaces and their tangents , volume=. Journal of Numerical Mathematics , publisher=. doi:10.1515/jnum-2013-0013 , number=

    work page doi:10.1515/jnum-2013-0013 2013

  7. [7]

    The length of vectors in representation spaces

    Kempf, George and Ness, Linda. The length of vectors in representation spaces. Algebraic Geometry. 1979

    1979

  8. [8]

    Unzerlegbare Darstellungen I

    Gabriel, Peter , journal =. Unzerlegbare Darstellungen I. , url =

  9. [9]

    , biburl =

    Schrijver, A. , biburl =

  10. [10]

    2021 , eprint=

    The capacity of quiver representations and Brascamp-Lieb constants , author=. 2021 , eprint=

    2021

  11. [11]

    2024 , eprint=

    Algorithmic aspects of semistability of quiver representations , author=. 2024 , eprint=

    2024

  12. [12]

    2016 , eprint=

    Degree bounds for semi-invariant rings of quivers , author=. 2016 , eprint=

    2016

  13. [13]

    Elements of the Representation Theory of Associative Algebras , volume=

    Simson, Daniel and Skowronski, Andrzej , year=. Elements of the Representation Theory of Associative Algebras , volume=. doi:10.1017/CBO9780511619403 , publisher=

    work page doi:10.1017/cbo9780511619403

  14. [14]

    Numerical Linear Algebra with Applications , year=

    Jordan's principal angles in complex vector spaces , author=. Numerical Linear Algebra with Applications , year=

  15. [15]

    G. D. Mostow , journal =. Self-Adjoint Groups , urldate =

  16. [16]

    On the orbit closure containment problem and slice rank of tensors , year =

    Bl\". On the orbit closure containment problem and slice rank of tensors , year =. Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms , pages =

  17. [17]

    2024 , eprint=

    Complexity theory of orbit closure intersection for tensors: reductions, completeness, and graph isomorphism hardness , author=. 2024 , eprint=

    2024

  18. [18]

    David Mumford , title =

  19. [19]

    Wallach , title =

    Nolan R. Wallach , title =

  20. [20]

    36th Computational Complexity Conference (CCC 2021) , pages =

    B\". 36th Computational Complexity Conference (CCC 2021) , pages =. 2021 , publisher =. doi:10.4230/LIPIcs.CCC.2021.32 , annote =

    work page doi:10.4230/lipics.ccc.2021.32 2021

  21. [21]

    Brooksbank and Eugene M

    Peter A. Brooksbank and Eugene M. Luks , keywords =. Testing isomorphism of modules , journal =. 2008 , note =. doi:https://doi.org/10.1016/j.jalgebra.2008.07.014 , url =

    work page doi:10.1016/j.jalgebra.2008.07.014 2008

  22. [22]

    On the orbit closure intersection problems for matrix tuples under conjugation and left-right actions , url =

    Gábor Ivanyos and Youming Qiao , booktitle =. On the orbit closure intersection problems for matrix tuples under conjugation and left-right actions , url =. doi:10.1137/1.9781611977554.ch158 , eprint =

    work page doi:10.1137/1.9781611977554.ch158

  23. [23]

    Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing , booktitle =

    Zeyuan Allen. Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing , booktitle =. 2018 , url =. doi:10.1145/3188745.3188942 , timestamp =

    work page doi:10.1145/3188745.3188942 2018

  24. [24]

    Degree bounds for semi-invariant rings of quivers , journal =

    Harm Derksen and Visu Makam , keywords =. Degree bounds for semi-invariant rings of quivers , journal =. 2018 , issn =. doi:https://doi.org/10.1016/j.jpaa.2017.12.007 , url =

    work page doi:10.1016/j.jpaa.2017.12.007 2018

  25. [25]

    1973 , publisher=

    The Representation Theory of Finite Graphs and Associated Algebras , author=. 1973 , publisher=

    1973

  26. [26]

    Mathematics of the USSR-Izvestiya , abstract =

    L A Nazarova , title =. Mathematics of the USSR-Izvestiya , abstract =. 1973 , month =. doi:10.1070/IM1973v007n04ABEH001975 , url =

    work page doi:10.1070/im1973v007n04abeh001975 1973

  27. [27]

    , journal =

    Hilbert, D. , journal =. Ueber die vollen Invariantensysteme , url =

  28. [28]

    and Shpilka, Amir

    Forbes, Michael A. and Shpilka, Amir. Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. 2013

    2013

  29. [29]

    On the orbit closure intersection problems for matrix tuples under conjugation and left-right actions , isbn =

    Ivanyos, Gábor and Qiao, Youming , year =. On the orbit closure intersection problems for matrix tuples under conjugation and left-right actions , isbn =