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When a quiver admits a height function, Euler characteristics of higher almost split complexes recover the truncated q-characters of standard modules in category C^{(1)}.

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2026-06-29 09:18 UTC pith:BD7WQRYE

load-bearing objection The paper builds a monoidal category R_Q and higher almost split complexes whose Euler characteristics recover truncated q-characters and type-A cluster characters.

arxiv 2605.28682 v1 pith:BD7WQRYE submitted 2026-05-27 math.RT math.COmath.CT

A higher homological approach to the q-characters of representations of quantum affine algebras

\'Elie Casbi This is my paper
classification math.RT math.COmath.CT
keywords q-charactersquantum affine algebrashigher almost split complexesmonoidal categoriescluster charactersexceptional sequencesHernandez-Leclerc categorypath algebras
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper constructs a monoidal category R_Q whose indecomposable objects are tensor products of finite-dimensional modules over the path algebra of an acyclic quiver Q without multiple edges. It establishes the existence and uniqueness up to homotopy of distinguished chain complexes called higher almost split complexes, which possess good homological properties preserved under tensoring by objects of R_Q. A key step is proving the existence of a family of complete exceptional sequences in the module category of the path algebra with many good properties. When Q admits a height function, the Euler characteristics of the images of these complexes under a certain additive functor equal the truncated q-characters of the standard modules in Hernandez-Leclerc's category C^{(1)}. For quivers of type A_n the same construction realizes the cluster characters of all cluster variables in the finite-type cluster algebra A_Q as Euler characteristics of certain chain complexes in R_Q.

Core claim

The central claim is that when Q admits a height function, the Euler characteristics of the images under a certain additive functor of the higher almost split complexes coincide with the truncated q-characters of the standard modules in Hernandez-Leclerc's category C^{(1)}. For the case where the underlying graph of Q is a Dynkin diagram of type A_n, the cluster characters of all cluster variables in the finite type cluster algebra A_Q are Euler characteristics of certain chain complexes in R_Q.

What carries the argument

Higher almost split complexes: distinguished chain complexes in the monoidal category R_Q that satisfy good homological properties and remain stable under tensoring by objects in R_Q.

Load-bearing premise

The existence of a family of complete exceptional sequences in the module category of the path algebra that satisfy many good properties.

What would settle it

For an explicit acyclic quiver Q of type A_2 that admits a height function, compute the Euler characteristic of the image under the additive functor of the associated higher almost split complex and check whether it equals the known truncated q-character of the corresponding standard module.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The truncated q-characters of standard modules admit a homological realization via Euler characteristics in R_Q.
  • Cluster characters of all cluster variables in type A_n arise as Euler characteristics of specific chain complexes in R_Q.
  • The higher almost split complexes are unique up to homotopy and their defining properties are preserved under tensor products.
  • The construction supplies a new link between the representation theory of acyclic quivers and the q-characters appearing in quantum affine algebra theory.

Where Pith is reading between the lines

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

  • The same homological machinery might furnish q-character interpretations for standard modules attached to quivers lacking a height function once suitable complexes are identified.
  • The exceptional sequences constructed as an intermediate step could be reused to produce similar higher almost split complexes in other monoidal categories built from quiver representations.
  • The approach suggests that Euler characteristics in R_Q may serve as a uniform source for both q-characters and cluster characters across different Dynkin types.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. For any acyclic quiver Q without multiple edges, the paper constructs a monoidal category R_Q whose indecomposable objects are tensor products of finite-dimensional kQ-modules. It shows existence and uniqueness up to homotopy of higher almost split chain complexes with good homological properties preserved under tensoring. A key step is establishing the existence of a family of complete exceptional sequences in mod kQ with many good properties. When Q admits a height function, the Euler characteristics of the images of these complexes under an additive functor coincide with the truncated q-characters of the standard modules in Hernandez-Leclerc's category C^(1). For type A_n, the cluster characters of all cluster variables in A_Q are interpreted as Euler characteristics of certain chain complexes in R_Q.

Significance. This work offers a higher homological perspective linking quiver module categories to q-characters of quantum affine algebra representations and to cluster characters in type A. The explicit construction of the complexes and the Euler characteristic equalities under the height function hypothesis provide a concrete bridge between homological algebra and these character theories; the family of complete exceptional sequences may also be of independent interest in representation theory of quivers.

minor comments (2)
  1. The abstract refers to an unspecified 'additive functor' whose images yield the Euler characteristics; this functor should be named and motivated in the introduction or §1.
  2. Notation for the higher almost split complexes and the monoidal structure on R_Q should be introduced with a brief table or diagram for clarity, especially when discussing preservation under tensoring.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, for highlighting its significance in providing a higher homological bridge between quiver representations, q-characters, and cluster characters, and for recommending minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No circularity: constructions and proofs are self-contained from quiver representation theory

full rationale

The paper constructs the monoidal category R_Q from finite-dimensional modules over kQ, establishes existence of complete exceptional sequences and higher almost split complexes as new objects, and proves Euler characteristic equalities to truncated q-characters under the height function hypothesis. These steps are forward derivations from standard quiver theory inputs; no equation reduces a claimed result to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The Hernandez-Leclerc reference is external and the type A_n cluster character interpretation follows from the constructions rather than presupposing the target equalities. This matches the default expectation of non-circularity for a paper whose central claims are existence and coincidence statements built from independent homological data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of complete exceptional sequences with good properties and on the quiver admitting a height function; these are domain assumptions in representation theory rather than new free parameters or invented entities.

axioms (2)
  • ad hoc to paper Existence of a family of complete exceptional sequences in mod kQ satisfying many good properties
    Explicitly called a crucial ingredient for constructing the higher almost split complexes.
  • domain assumption Q admits a height function
    Required for the Euler characteristic to coincide with truncated q-characters.

pith-pipeline@v0.9.1-grok · 5755 in / 1381 out tokens · 30836 ms · 2026-06-29T09:18:39.049560+00:00 · methodology

0 comments
read the original abstract

For any acyclic quiver $Q$ without multiple edges, we construct a monoidal category $\mathcal{R}_Q$ whose indecomposable objects are tensor products (over the base field) of finite-dimensional modules over the path algebra of $Q$. We show the existence and uniqueness up to homotopy of certain distinguished chain complexes satisfying good homological properties (higher almost split complexes) preserved under tensoring by objects in $\mathcal{R}_Q$. As a crucial ingredient for this construction, we establish the existence of a family of complete exceptional sequences in $\mathrm{mod}\,\mathbf{k}Q$ satisfying many good properties, which we believe might be of independent interest. We then prove that when $Q$ admits a height function, the Euler characteristics of (the images under certain additive functor of) these complexes coincide with the truncated $q$-characters of the standard modules in Hernandez-Leclerc's category $\mathcal{C}^{(1)}$. Applying our results to the case where the underlying graph of $Q$ is a Dynkin diagram of type $A_n, n \geq 1$, we also interpret the cluster characters of all cluster variables in the finite type cluster algebra $\mathcal{A}_Q$ as Euler characteristics of certain chain complexes in $\mathcal{R}_Q$.

Figures

Figures reproduced from arXiv: 2605.28682 by \'Elie Casbi.

Figure 1
Figure 1. Figure 1: Illustration of (4.7) for a quiver of type A3: the union of the hammock multisets of τy and y (in red) can be decomposed as the union of those of x and τz (in blue), together with the set containing only τy and its shift τ −1 y (in green). This will follow from the fact that x ι−→ L x→z z π−→ τ −1x → Σx is an Auslander-Reiten triangle in Db (mod kQ). Indeed, let y ∈ IQ and assume y is not isomorphic neithe… view at source ↗
Figure 2
Figure 2. Figure 2: The chain complex C•(Mβ) (on the left) and the graph displaying the classes in K0(HQ) of the images of each object under the functor DQ (on the right). monomials, and that their truncated q-characters can be identified (up to a mild change of variables) with their cluster expansion with respect to an appropriately chosen initial seed. Therefore in what follows we will essentially work exclusively using the… view at source ↗

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