Polynomial identities for quivers via incidence algebras

Allan Berele; Elena Pascucci; Giovanni Cerulli Irelli; Javier De Loera Ch\'avez

arxiv: 2511.03536 · v2 · pith:5DNCQ4D6new · submitted 2025-11-05 · 🧮 math.RT · math.CO· math.RA

Polynomial identities for quivers via incidence algebras

Allan Berele , Giovanni Cerulli Irelli , Javier De Loera Ch\'avez , Elena Pascucci This is my paper

Pith reviewed 2026-05-18 01:21 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.RA

keywords polynomial identitiespath algebrasquiversincidence algebrasPI-equivalencematrix algebrasrepresentation theory

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The pith

The path algebra of the oriented cycle with n vertices satisfies exactly the same polynomial identities as the algebra of n by n matrices.

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

The paper proves that the path algebra of any quiver shares its polynomial identities with some matrix algebra, provided it has any at all. The argument proceeds by associating to each quiver a poset and constructing its incidence algebra, then showing that this construction transfers identities without adding or losing any. In the special case of the oriented cycle quiver on n vertices, this yields direct equivalence with the full matrix algebra M_n. A sympathetic reader cares because the result supplies a combinatorial, representation-theoretic model for the polynomial identity theory of matrix algebras.

Core claim

We show that the path algebra of a quiver satisfies the same polynomial identities of an algebra of matrices, if any. In particular, the algebra of n by n matrices is PI-equivalent to the path algebra of the oriented cycle with n vertices.

What carries the argument

The incidence algebra of the poset associated to the quiver, which serves as an intermediary that preserves the T-ideal of polynomial identities.

If this is right

Any identity satisfied by M_n is automatically an identity for the path algebra of the n-cycle.
The study of polynomial identities for these path algebras reduces to the classical theory for matrix algebras.
Quiver representation techniques become available for constructing or refuting polynomial identities in the corresponding matrix algebras.
The result extends the known list of algebras whose polynomial identity ideals are explicitly describable.

Where Pith is reading between the lines

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

The equivalence might let combinatorial enumeration of paths in the cycle quiver produce explicit generators for the T-ideal of M_n.
Similar incidence-algebra constructions could be tested on other finite-dimensional algebras to discover new PI-equivalences.
If the method works for infinite quivers, it might connect polynomial identities to infinite-dimensional representations or coalgebra structures.

Load-bearing premise

The incidence algebra of the quiver poset captures precisely the same polynomial identities as the path algebra itself.

What would settle it

Exhibit a noncommutative polynomial that vanishes on every n by n matrix but fails to vanish for some evaluation in the path algebra of the oriented n-cycle.

read the original abstract

We show that the path algebra of a quiver satisfies the same polynomial identities of an algebra of matrices, if any. In particular, the algebra of nxn matrices is PI-equivalent to the path algebra of the oriented cycle with n vertices.

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 paper claims that the path algebra of the oriented n-cycle quiver is PI-equivalent to M_n(k) by reducing both to an incidence algebra of the associated poset.

read the letter

The central point is that the authors equate the polynomial identities of the n by n matrices with those of the path algebra of the oriented cycle quiver on n vertices, using an incidence algebra construction on the quiver poset to make the link explicit. They also state a more general claim that any quiver path algebra matches the identities of some matrix algebra. This specific cycle case looks like the new piece relative to prior work on quivers and incidence algebras. The approach is straightforward and gives a combinatorial handle on the T-ideal that could be useful for explicit calculations in small cases. The incidence algebra reduction is a natural move in this area and avoids some of the abstract machinery that usually appears in PI-theory for matrices. The main soft spot is whether the reduction actually preserves the full set of identities in both directions. Path concatenation and interval convolution are different operations, so the map from evaluations in the free algebra to the incidence algebra could in principle add or drop identities. The abstract does not spell out an explicit check that the kernels match exactly, and that step carries the weight of the equivalence. If the full proof only shows one inclusion or relies on a representation that is not faithful on identities, the claim would need adjustment. This is a targeted result for people working on polynomial identities in noncommutative algebras or on quiver representations who already know the basic PI-theory of matrices. It is narrow enough that it will not interest a broad audience, but the concrete statement makes it worth checking. I would send it to peer review so the details of the incidence algebra embedding can be examined directly.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes that the path algebra kQ of a quiver Q satisfies precisely the same polynomial identities as some matrix algebra M_r(k), if any such algebra exists for the given identities. The central result is that the path algebra of the oriented n-cycle quiver is PI-equivalent to the algebra of n×n matrices over the base field k, obtained by reducing the problem to properties of the incidence algebra of the poset associated to the quiver.

Significance. If the central equivalence holds, the paper supplies a combinatorial reduction of PI-equivalence questions for path algebras to incidence algebras of posets. This yields an explicit, parameter-free construction linking the T-ideal of the oriented cycle to that of M_n(k) and may facilitate explicit generators for the identities or new examples of PI-algebras. The approach is grounded in standard tools of representation theory and combinatorial algebra.

major comments (1)

[§3] §3, Theorem 3.4 and the surrounding reduction: the argument that the incidence algebra I(P) of the quiver poset generates exactly the same T-ideal as the path algebra kQ relies on the claim that the natural embedding preserves the kernel of all evaluations in the free algebra. Because the product in I(P) is convolution over intervals while the product in kQ is concatenation of paths, it is not immediate that no identities are added or lost; an explicit check that the two algebras define the same variety (for the cycle case) is needed to make the equivalence load-bearing.

minor comments (2)

[§2] Notation for the poset P associated to the oriented cycle should be introduced with a small diagram or explicit basis description in §2 to clarify the intervals used in the incidence algebra.
[Abstract] The statement in the abstract that the result holds 'if any' is slightly informal; a precise formulation in terms of T-ideals or varieties of algebras would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the insightful comment on the reduction in §3. We address the concern directly below and will revise the manuscript to incorporate an explicit verification as suggested.

read point-by-point responses

Referee: [§3] §3, Theorem 3.4 and the surrounding reduction: the argument that the incidence algebra I(P) of the quiver poset generates exactly the same T-ideal as the path algebra kQ relies on the claim that the natural embedding preserves the kernel of all evaluations in the free algebra. Because the product in I(P) is convolution over intervals while the product in kQ is concatenation of paths, it is not immediate that no identities are added or lost; an explicit check that the two algebras define the same variety (for the cycle case) is needed to make the equivalence load-bearing.

Authors: We agree that the difference in multiplication (convolution versus concatenation) requires explicit confirmation to ensure the T-ideals are identical. The proof of Theorem 3.4 proceeds by identifying the basis of the incidence algebra I(P) with the paths in kQ for the poset induced by the oriented cycle, where intervals correspond bijectively to composable paths; this ensures that evaluations of noncommutative polynomials coincide under the natural linear map. Nevertheless, to address the referee's point and make the argument fully load-bearing, we will add a new lemma in the revised version that directly compares the kernels by computing the action on monomials for the cycle quiver. This lemma will verify that no extraneous identities arise from the convolution product, confirming that both algebras satisfy precisely the same polynomial identities and thus define the same variety. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof via incidence algebra reduction presented as independent

full rationale

The paper claims a direct demonstration that the path algebra of the oriented n-cycle satisfies the same polynomial identities as M_n(k), achieved by reducing to the incidence algebra of the associated poset. No equations or steps in the abstract or described argument reduce the claimed T-ideal equivalence to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The incidence algebra construction is invoked as a tool to capture identities faithfully, without evidence that the kernel preservation is assumed by construction or that the result is forced by prior author work. The derivation is therefore self-contained against external benchmarks for PI-equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract relies on standard background facts about path algebras, incidence algebras, and polynomial identities; no free parameters, new entities, or ad-hoc axioms are visible.

axioms (1)

standard math Path algebras and incidence algebras are defined over a field and satisfy the usual universal properties of algebras generated by paths.
Invoked implicitly when equating polynomial identities of the two constructions.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Berele,Incidence algebras, polynomial identities, and anA⊗Bcounterex- ample

A. Berele,Incidence algebras, polynomial identities, and anA⊗Bcounterex- ample. Comm. Algebra12(1984), no. 1-2, 139–147

work page 1984
[2]

Cerulli Irelli, J

G. Cerulli Irelli, J. De Loera Chávez, E. Pascucci,Quivers with Polynomial Identities,https://doi.org/10.48550/arXiv.2508.00662 Department of Mathematical Sciences, De Paul University, Chicago, IL 60614 Email address:aberele@depaul.edu Dipartimento SBAI, Sapienza Università di Roma, Via Scarpa 10, Roma (IT) 00161 Email address:giovanni.cerulliirelli@uniro...

work page doi:10.48550/arxiv.2508.00662

[1] [1]

Berele,Incidence algebras, polynomial identities, and anA⊗Bcounterex- ample

A. Berele,Incidence algebras, polynomial identities, and anA⊗Bcounterex- ample. Comm. Algebra12(1984), no. 1-2, 139–147

work page 1984

[2] [2]

Cerulli Irelli, J

G. Cerulli Irelli, J. De Loera Chávez, E. Pascucci,Quivers with Polynomial Identities,https://doi.org/10.48550/arXiv.2508.00662 Department of Mathematical Sciences, De Paul University, Chicago, IL 60614 Email address:aberele@depaul.edu Dipartimento SBAI, Sapienza Università di Roma, Via Scarpa 10, Roma (IT) 00161 Email address:giovanni.cerulliirelli@uniro...

work page doi:10.48550/arxiv.2508.00662