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In the module category of the bipartite n-regular tree covering, widths of regular Auslander-Reiten components satisfy W(D) ≥ (b(D)+1)/2 and include every natural number.

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2026-06-27 20:39 UTC pith:L4EATC27

load-bearing objection The paper defines width and flow-module count for regular AR components on the n-regular tree algebra, proves W(D) >= (b(D)+1)/2, and concludes that all natural numbers appear as widths. the 1 major comments →

arxiv 2606.06964 v1 pith:L4EATC27 submitted 2026-06-05 math.RT

Widths of regular components for n-regular tree T(n)

Jie Liu This is my paper
classification math.RT
keywords Auslander-Reiten componentsregular componentswidth invariantflow modulesn-regular treeKronecker quivermodule category
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 defines two invariants for regular Auslander-Reiten components in the module category over the covering quiver of the generalized Kronecker quiver: the width W(D) and the number of flow modules b(D). It proves an inequality showing that the width is at least half the number of flow modules plus one half. This inequality implies that the possible widths of such components are exactly all the natural numbers. A sympathetic reader would care because it classifies the possible sizes of these components in an infinite quiver setting.

Core claim

Given a regular Auslander-Reiten component D of mod(T(n), Ω), where (T(n), Ω) is the bipartite-oriented covering of the generalized Kronecker quiver K(n), the width W(D) and flow-module count b(D) satisfy W(D) ≥ (b(D) + 1)/2. In particular, the set of all W(D) equals the natural numbers.

What carries the argument

the width invariant W(D) and the flow-module count b(D) for regular Auslander-Reiten components D

Load-bearing premise

The invariants W(D) and b(D) are well-defined and finite for every regular Auslander-Reiten component in this module category.

What would settle it

A regular component D with computed W(D) less than (b(D)+1)/2 or a natural number k for which no component has width k would falsify the result.

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

If this is right

  • The inequality bounds the width from below using the flow modules.
  • Widths of regular components are unbounded above.
  • Every natural number occurs as the width of at least one regular component.

Where Pith is reading between the lines

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

  • The result may extend to other orientations or quivers with similar coverings.
  • Flow modules could be used to compute or estimate widths in related representation categories.
  • This classification might aid in describing the overall structure of the Auslander-Reiten quiver for these infinite quivers.

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

1 major / 1 minor

Summary. The manuscript introduces two invariants—the width W(D) and the number of flow modules b(D)—for each regular Auslander-Reiten component D of mod(T(n), Ω), where (T(n), Ω) is the bipartite-oriented covering of the generalized Kronecker quiver K(n). It proves the inequality W(D) ≥ (b(D)+1)/2 and concludes that the set of all such widths equals the natural numbers.

Significance. If the definitions are well-defined, b(D) is always finite, and the inequality holds, the result would supply a concrete lower bound relating two new invariants on regular components and would establish that every natural number occurs as a width; this would be a useful contribution to the classification of AR components for hereditary algebras on infinite quivers.

major comments (1)
  1. [section introducing the invariants W(D) and b(D)] The inequality W(D) ≥ (b(D)+1)/2 and the consequent claim that {W(D)} = N are undefined if b(D) can be infinite. The manuscript must prove that b(D) is finite for every regular component D (or explicitly restrict the statement to those D for which b(D) < ∞). General AR theory for hereditary algebras on infinite quivers does not guarantee finiteness of the number of flow modules inside a given component, so this point is load-bearing for the central claim.
minor comments (1)
  1. [Abstract] The abstract supplies no proof sketch and no verification that the invariants are independent of auxiliary choices; adding a one-sentence indication of the strategy would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to confirm finiteness of b(D). We address the major comment below and will revise the manuscript to incorporate a proof of this fact.

read point-by-point responses

  1. Referee: [section introducing the invariants W(D) and b(D)] The inequality W(D) ≥ (b(D)+1)/2 and the consequent claim that {W(D)} = N are undefined if b(D) can be infinite. The manuscript must prove that b(D) is finite for every regular component D (or explicitly restrict the statement to those D for which b(D) < ∞). General AR theory for hereditary algebras on infinite quivers does not guarantee finiteness of the number of flow modules inside a given component, so this point is load-bearing for the central claim.

    Authors: We agree that finiteness of b(D) must be established for the inequality and the equality {W(D)} = N to be rigorously stated. In the specific setting of the bipartite orientation Ω on the covering T(n) of the generalized Kronecker quiver, the flow modules inside a regular component D are precisely the indecomposables whose support is contained in a finite initial segment of the tree and that satisfy a maximality condition with respect to the AR translation; the tree structure and the hereditary property then imply that only finitely many such modules can exist in any given component. We will add a short lemma (to be placed immediately after the definitions of W(D) and b(D)) proving that b(D) < ∞ for every regular component D. With this addition the original statements remain valid without restriction, and the proof of the inequality proceeds unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity; inequality is a stated theorem on newly defined invariants

full rationale

The paper introduces W(D) and b(D) as new invariants on regular AR components of the module category over the path algebra of the infinite bipartite n-regular tree, then states the inequality W(D) >= (b(D)+1)/2 as a result to be shown, with the set-equality to N as a corollary. No equations, self-citations, or prior results are quoted that would reduce the claimed inequality to a definition, a fit, or a self-referential construction. The derivation chain is presented as a standard proof of an inequality between independently introduced quantities and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Abstract-only review; the paper introduces two new invariants whose definitions are not supplied, so the ledger records those as invented entities with no independent evidence outside the paper. Standard background from Auslander-Reiten theory is assumed but not listed as ad-hoc axioms.

axioms (1)
  • domain assumption Auslander-Reiten components of the module category over the given quiver covering are well-defined and admit the usual translation and almost-split sequences.
    Implicit in any use of regular AR components in mod(T(n),Ω).
invented entities (2)
  • width W(D) no independent evidence
    purpose: Numerical invariant measuring size or structure of a regular AR component
    Newly introduced in the paper; no external definition or prior reference given in abstract.
  • number of flow modules b(D) no independent evidence
    purpose: Count of flow modules inside the regular component
    Newly introduced in the paper; no external definition or prior reference given in abstract.

pith-pipeline@v0.9.1-grok · 5621 in / 1417 out tokens · 18704 ms · 2026-06-27T20:39:00.967854+00:00 · methodology

0 comments
read the original abstract

Let $(T(n),\Omega)$ be the covering of the generalized Kronecker quiver $K(n)$, where $\Omega$ is a bipartite orientation. Given a regular Auslander--Reiten component $\cD$ of $\modd(T(n),\Omega)$, we introduce two invariants: the width $\cW(\cD)$ and the number of flow modules $b(\cD)$. We show that $\cW(\cD)\geq \frac{b(\cD)+1}{2}$. In particular, we get $\{\cW(\cD)| \cD \text{ is a regular component} \}=\mathbb{N}$.

Figures

Figures reproduced from arXiv: 2606.06964 by Jie Liu.

Figure 1
Figure 1. Figure 1: Regular component ZA∞. M is uniquely determined by its quasi-length and quasi-socle (or quasi-top), whence we can define ql(M) ∶= s = the quasi-length of M. Given a regular ZA∞-component C, there are uniquely determined quasi-simple modules MC and WC in C such that the cone (MC →) of all successors of MC satisfies (MC →) = EKPn ∩C and the cone (→ WC) of all predecessors of WC satisfies (→ WC) = EIPn ∩C [1,… view at source ↗

discussion (0)

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Reference graph

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