The Deligne-Simpson Problem

Andrew Hubery; William Crawley-Boevey

arxiv: 2509.11998 · v5 · submitted 2025-09-15 · 🧮 math.RA

The Deligne-Simpson Problem

William Crawley-Boevey , Andrew Hubery This is my paper

Pith reviewed 2026-05-18 16:52 UTC · model grok-4.3

classification 🧮 math.RA

keywords Deligne-Simpson problemroot systemssimilarity classesinvertible matricesquiver representationsmatrix productsinvariant subspaces

0 comments

The pith

The Deligne-Simpson problem admits a solution precisely when its associated root system satisfies the conjectured condition.

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

The paper completes the proof that, given similarity classes of invertible matrices, matrices from those classes can be chosen so their product is the identity and they share no common invariant subspace exactly when a root system built from the classes meets a specific condition. One direction of this equivalence was already known from earlier joint work; this paper establishes the converse. A sympathetic reader cares because the result supplies a complete, checkable criterion that turns an existence question about matrices into a question about root systems.

Core claim

Given k similarity classes of invertible matrices, the Deligne-Simpson problem asks whether matrices exist in these classes whose product is the identity and that have no common invariant subspace. The paper proves this is possible if and only if the root system associated to the classes satisfies the condition previously conjectured by the first author, thereby confirming the full conjecture.

What carries the argument

The root system associated to the similarity classes, which translates the matrix existence question into a combinatorial condition on the root system.

If this is right

Solvability can now be decided by inspecting the root system rather than by searching for the matrices directly.
The geometric link to quiver representations becomes a two-way correspondence for deciding irreducibility and product-one conditions.
The full equivalence allows systematic classification of solvable instances for any number of similarity classes.

Where Pith is reading between the lines

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

The criterion may support computational checks for small k by enumerating roots and verifying the condition.
Analogous root-system tests might apply to related questions about matrix tuples with other algebraic constraints.
The translation between matrices and quiver representations invites study of the same problem over fields of positive characteristic.

Load-bearing premise

The root system is built from the similarity classes in the specific way defined in the earlier joint work with Shaw, and the base field is algebraically closed of characteristic zero.

What would settle it

An explicit set of similarity classes for which the root system satisfies the condition but no such matrix tuple exists, or for which the condition fails yet the matrices can still be found.

read the original abstract

Given k similarity classes of invertible matrices, the Deligne-Simpson problem asks to determine whether or not one can find matrices in these classes whose product is the identity and with no common invariant subspace. The first author conjectured an answer in terms of an associated root system, and proved one implication in joint work with Shaw. In this paper we prove the other implication, thus confirming the conjecture.

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 finishes the Deligne-Simpson conjecture by proving the missing sufficiency direction for the root-system condition.

read the letter

The main point is that this paper proves the converse implication of the Deligne-Simpson conjecture. Given similarity classes of invertible matrices, the problem asks whether matrices from those classes exist whose product is the identity and which share no common invariant subspace. Crawley-Boevey conjectured an answer in terms of an associated root system, and the necessity direction was already shown with Shaw. This work establishes sufficiency, so the full if-and-only-if statement now holds under the standard hypotheses of an algebraically closed field of characteristic zero.

Referee Report

0 major / 2 minor

Summary. The manuscript proves the missing implication of the Deligne-Simpson conjecture: given similarity classes of invertible matrices over an algebraically closed field of characteristic zero, a tuple of matrices from these classes exists with product equal to the identity and no common invariant subspace if and only if the associated root system satisfies the condition conjectured by the first author. The argument establishes the existence direction by invoking the geometric translation to quiver representations defined in the authors' prior joint work with Shaw.

Significance. If the result holds, the paper completes the full characterization of the Deligne-Simpson problem via a root-system criterion. This advances the understanding of irreducible tuples of matrices with prescribed conjugacy classes and strengthens the link to quiver representations, offering a combinatorial test that may support further explicit computations or extensions in representation theory of algebraic groups.

minor comments (2)

The introduction would benefit from an explicit statement of the main theorem (including the precise root-system condition) immediately after the conjecture is recalled, to orient readers before the technical sections.
Notation for the root system and the associated quiver could be cross-referenced more explicitly to the earlier joint work with Shaw when the geometric translation is invoked.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance. We appreciate the recognition that the work completes the characterization of the Deligne-Simpson problem via the root-system criterion.

Circularity Check

0 steps flagged

Completes external conjecture; no load-bearing self-referential reduction

full rationale

The paper proves the missing implication of the Deligne-Simpson conjecture originally stated by the first author. It relies on the geometric translation to quiver representations and the root-system construction defined in prior joint work with Shaw, but these are external references rather than internal redefinitions or fitted parameters renamed as predictions. The central argument establishes existence under the root-system condition without reducing the result to a self-citation chain or ansatz smuggled from the authors' own prior definitions. The derivation remains self-contained against the standing hypotheses of an algebraically closed field of characteristic zero.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior construction of the root system from the similarity classes and on standard facts from algebraic geometry of quiver moduli spaces; no new free parameters or invented entities are introduced in this paper.

axioms (2)

domain assumption The root system associated to the similarity classes is defined exactly as in the earlier joint work with Shaw.
Invoked when translating the matrix product condition into a statement about roots.
domain assumption The base field is algebraically closed of characteristic zero.
Required for the geometric arguments using quiver representations.

pith-pipeline@v0.9.0 · 5576 in / 1298 out tokens · 35737 ms · 2026-05-18T16:52:22.519377+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: irreducible solution iff α=α_C positive root, ξ[α]=1 and p_w(α)>p_w(β)+... for nontrivial decompositions with ξ[β]=1 etc. (proved via Λ_q(Q_w)-modules and cases (I)-(III) in Thm 7.4)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Root lattice Γ_w, Euler form ⟨−,−⟩_Q, p_Q(α)=1−q_Q(α); star quiver Q_w with vertices [i,j]

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.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Crawley-Boevey, Geometry of the moment map for representations of quiv- ers, Compositio Math

W. Crawley-Boevey, Geometry of the moment map for representations of quiv- ers, Compositio Math. 126 (2001), 257–293

work page 2001
[2]

Crawley-Boevey, Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the iden- tity, Publ

W. Crawley-Boevey, Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the iden- tity, Publ. Math. Inst. Hautes Etudes Sci. 100 (2004), 171–207

work page 2004
[3]

Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne-Simpson problem, International Congress of Mathematicians

W. Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne-Simpson problem, International Congress of Mathematicians. Vol. II, 117–129, Eur. Math. Soc., Z¨ urich, 2006

work page 2006
[4]

Crawley-Boevey, Kac’s Theorem for weighted projective lines, J

W. Crawley-Boevey, Kac’s Theorem for weighted projective lines, J. Eur. Math. Soc. (JEMS) 12 (2010), 1331–1345

work page 2010
[5]

Crawley-Boevey, Connections for weighted projective lines, J

W. Crawley-Boevey, Connections for weighted projective lines, J. Pure Appl. Algebra, 215 (2011), 35–43

work page 2011
[6]

Crawley-Boevey and A

W. Crawley-Boevey and A. Hubery, A new approach to simple modules for preprojective algebras, Algebras and Representation Theory 23 (2020), 1849- 1860. 24 WILLIAM CRA WLEY-BOEVEY AND ANDREW HUBERY

work page 2020
[7]

Crawley-Boevey and P

W. Crawley-Boevey and P. Shaw, Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem, Adv. Math. 201 (2006), 180– 208

work page 2006
[8]

Geigle and H

W. Geigle and H. Lenzing, A class of weighted projective curves arising in rep- resentation theory of finite-dimensional algebras, in: Singularities, representa- tion of algebras, and vector bundles (Proc. Lambrecht, 1985), G.-M. Greuel and G. Trautmann (eds.), 265–297, Lecture Notes in Math., 1273, Springer, Berlin, 1987

work page 1985
[9]

Geigle and H

W. Geigle and H. Lenzing, Perpendicular categories with applications to rep- resentations and sheaves, J. Algebra 144 (1991), 273–343

work page 1991
[10]

A. Hubery, Connections on sheaves on weighted projective lines, and an appli- cation to the Deligne-Simpson Problem, Habilitationshrift, Fakult¨ at f¨ ur Math- ematik, Universit¨ at Bielefeld, March 2023

work page 2023
[11]

H¨ ubner and H

T. H¨ ubner and H. Lenzing, Categories perpendicular to exceptional bundles, manuscript 1993

work page 1993
[12]

V. G. Kac, Root systems, representations of quivers and invariant theory, in: Invariant theory (Proc. Montecatini, 1982), F. Gherardelli (ed.), 74–108, Lec- ture Notes in Math., 996, Springer, Berlin, 1983

work page 1982
[13]

V. P. Kostov, The Deligne-Simpson problem—a survey,J. Algebra281(2004), 83–108

work page 2004
[14]

Lenzing, Representations of finite-dimensional algebras and singularity the- ory, in: Trends in ring theory (Miskolc, 1996), 71–97, Canadian Math

H. Lenzing, Representations of finite-dimensional algebras and singularity the- ory, in: Trends in ring theory (Miskolc, 1996), 71–97, Canadian Math. Soc. Conf. Proc., 22, Amer. Math. Soc., Providence, RI, 1998

work page 1996
[15]

Lenzing and H

H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, in: Representations of algebras (Ottawa, ON, 1992), 313–337, CMS Conf. Proc., 14, Amer. Math. Soc., Prov- idence, RI, 1993

work page 1992
[16]

Mihai, Sur les connexions m´ eromorphes, Rev

A. Mihai, Sur les connexions m´ eromorphes, Rev. Roum. Math. Pures et Appl., 23(1978), 215–232

work page 1978
[17]

Schiffmann, Noncommutative projective curves and quantum loop algebras, Duke Math

O. Schiffmann, Noncommutative projective curves and quantum loop algebras, Duke Math. J. 121 (2004), 113–168

work page 2004
[18]

Shu, The tame Deligne-Simpson problem, arxiv:2509.11841 [math.RT]

C. Shu, The tame Deligne-Simpson problem, arxiv:2509.11841 [math.RT]. F akult¨at f ¨ur Mathematik, Universit ¨at Bielefeld, 33501 Bielefeld, Germany Email address:wcrawley@math.uni-bielefeld.de Email address:hubery@math.uni-bielefeld.de

work page arXiv

[1] [1]

Crawley-Boevey, Geometry of the moment map for representations of quiv- ers, Compositio Math

W. Crawley-Boevey, Geometry of the moment map for representations of quiv- ers, Compositio Math. 126 (2001), 257–293

work page 2001

[2] [2]

Crawley-Boevey, Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the iden- tity, Publ

W. Crawley-Boevey, Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the iden- tity, Publ. Math. Inst. Hautes Etudes Sci. 100 (2004), 171–207

work page 2004

[3] [3]

Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne-Simpson problem, International Congress of Mathematicians

W. Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne-Simpson problem, International Congress of Mathematicians. Vol. II, 117–129, Eur. Math. Soc., Z¨ urich, 2006

work page 2006

[4] [4]

Crawley-Boevey, Kac’s Theorem for weighted projective lines, J

W. Crawley-Boevey, Kac’s Theorem for weighted projective lines, J. Eur. Math. Soc. (JEMS) 12 (2010), 1331–1345

work page 2010

[5] [5]

Crawley-Boevey, Connections for weighted projective lines, J

W. Crawley-Boevey, Connections for weighted projective lines, J. Pure Appl. Algebra, 215 (2011), 35–43

work page 2011

[6] [6]

Crawley-Boevey and A

W. Crawley-Boevey and A. Hubery, A new approach to simple modules for preprojective algebras, Algebras and Representation Theory 23 (2020), 1849- 1860. 24 WILLIAM CRA WLEY-BOEVEY AND ANDREW HUBERY

work page 2020

[7] [7]

Crawley-Boevey and P

W. Crawley-Boevey and P. Shaw, Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem, Adv. Math. 201 (2006), 180– 208

work page 2006

[8] [8]

Geigle and H

W. Geigle and H. Lenzing, A class of weighted projective curves arising in rep- resentation theory of finite-dimensional algebras, in: Singularities, representa- tion of algebras, and vector bundles (Proc. Lambrecht, 1985), G.-M. Greuel and G. Trautmann (eds.), 265–297, Lecture Notes in Math., 1273, Springer, Berlin, 1987

work page 1985

[9] [9]

Geigle and H

W. Geigle and H. Lenzing, Perpendicular categories with applications to rep- resentations and sheaves, J. Algebra 144 (1991), 273–343

work page 1991

[10] [10]

A. Hubery, Connections on sheaves on weighted projective lines, and an appli- cation to the Deligne-Simpson Problem, Habilitationshrift, Fakult¨ at f¨ ur Math- ematik, Universit¨ at Bielefeld, March 2023

work page 2023

[11] [11]

H¨ ubner and H

T. H¨ ubner and H. Lenzing, Categories perpendicular to exceptional bundles, manuscript 1993

work page 1993

[12] [12]

V. G. Kac, Root systems, representations of quivers and invariant theory, in: Invariant theory (Proc. Montecatini, 1982), F. Gherardelli (ed.), 74–108, Lec- ture Notes in Math., 996, Springer, Berlin, 1983

work page 1982

[13] [13]

V. P. Kostov, The Deligne-Simpson problem—a survey,J. Algebra281(2004), 83–108

work page 2004

[14] [14]

Lenzing, Representations of finite-dimensional algebras and singularity the- ory, in: Trends in ring theory (Miskolc, 1996), 71–97, Canadian Math

H. Lenzing, Representations of finite-dimensional algebras and singularity the- ory, in: Trends in ring theory (Miskolc, 1996), 71–97, Canadian Math. Soc. Conf. Proc., 22, Amer. Math. Soc., Providence, RI, 1998

work page 1996

[15] [15]

Lenzing and H

H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, in: Representations of algebras (Ottawa, ON, 1992), 313–337, CMS Conf. Proc., 14, Amer. Math. Soc., Prov- idence, RI, 1993

work page 1992

[16] [16]

Mihai, Sur les connexions m´ eromorphes, Rev

A. Mihai, Sur les connexions m´ eromorphes, Rev. Roum. Math. Pures et Appl., 23(1978), 215–232

work page 1978

[17] [17]

Schiffmann, Noncommutative projective curves and quantum loop algebras, Duke Math

O. Schiffmann, Noncommutative projective curves and quantum loop algebras, Duke Math. J. 121 (2004), 113–168

work page 2004

[18] [18]

Shu, The tame Deligne-Simpson problem, arxiv:2509.11841 [math.RT]

C. Shu, The tame Deligne-Simpson problem, arxiv:2509.11841 [math.RT]. F akult¨at f ¨ur Mathematik, Universit ¨at Bielefeld, 33501 Bielefeld, Germany Email address:wcrawley@math.uni-bielefeld.de Email address:hubery@math.uni-bielefeld.de

work page arXiv