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arxiv: 2406.03604 · v3 · pith:3YCYYSEPnew · submitted 2024-06-05 · 🧮 math.RT · math.CO· math.RA

Cyclically ordered quivers

Sergey Fomin , Scott Neville This is my paper

Pith reviewed 2026-05-24 00:21 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.RA MSC 16G2013F60
keywords quiversmutationcyclic orderinginvariantscluster algebrasrepresentation theory
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The pith

A cyclic ordering on the vertices of a quiver produces new powerful mutation invariants.

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

The paper introduces cyclically ordered quivers by adding a cyclic ordering to the vertices of an ordinary quiver. This extra structure, which appears in many applications, generates mutation invariants that are stronger than those available from the quiver alone. These invariants remain unchanged when the quiver undergoes mutation while respecting the cyclic order. The construction refines the usual mutation equivalence relation and supplies new tools for distinguishing quivers that would otherwise be considered equivalent.

Core claim

A cyclically ordered quiver is a quiver together with a cyclic ordering of its vertices; this structure gives rise to new powerful mutation invariants that are preserved under the natural extension of quiver mutation to the ordered setting.

What carries the argument

The cyclic ordering of vertices, which augments the quiver to define additional invariants preserved by mutation.

Load-bearing premise

The cyclic ordering of vertices is a structure that naturally arises in many important applications.

What would settle it

An explicit pair of cyclically ordered quivers related by a sequence of mutations whose associated invariants differ.

Figures

Figures reproduced from arXiv: 2406.03604 by Scott Neville, Sergey Fomin.

Figure 1
Figure 1. Figure 1: The hierarchy of equivalence classes, orbits, and invariants. Here [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An acyclic and complete (but not abundant) quiver on 4 vertices. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: a tree quiver. Right: a disconnected quiver. Neither quiver is complete. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An acyclic 3-vertex quiver. a b c x y −z [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: The six COQs whose underlying quiver is the 4-cycle [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Four COQs related by a sequence of wiggles. Also, the first COQ is related [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A 4-vertex acyclic quiver. Remark 4.16. There are known algorithms for testing whether two given matrices in GLn(Z) are conjugate to each other (over Z). We will discuss this topic, and provide references, in Section 7, see Remark 7.6 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: In this COQ, the vertices g, h, i, and j are proper but k and ℓ are not. Remark 5.3. Properness of an individual vertex in a COQ is not preserved under wiggles: a wiggle may transform a proper vertex into a non-proper one. To see it, suppose that two vertices i and k are adjacent in the cyclic ordering but not connected by an arrow in Q. If an oriented path i → j → k makes a right turn at j, then the same … view at source ↗
Figure 10
Figure 10. Figure 10: The proper mutation class of a COQ of type [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Proper mutation classes of COQs of type A3, considered up to relabeling. Each box contains a wiggle equivalence class. Double-sided arrows represent proper mutations. The red vertices are not proper [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Proper mutation classes of COQs of type A4, considered up to relabeling, taking the opposites (cf. Definition 6.7), and wiggle equivalence. Red vertices are not proper. Example 6.10. Consider the oriented 4-cycle quiver Q of type D4, with arrows a → b → c → d → a. This quiver has three wiggle equivalence classes of cyclic orderings, cf. Example 2.7, with representatives σ1 = (a, b, c, d), σ2 = (a, b, d, c… view at source ↗
Figure 13
Figure 13. Figure 13: Every vertex in each of these quivers is proper. [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: Proper mutation classes of COQs of type A˜(3, 1), considered up to rela￾beling, opposites, and wiggle equivalence. The red vertecies are not proper [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: A 3-cycle quiver with multiplicities 2, m, m. Remark 7.10. The above trick is not guaranteed to always work to establish non￾conjugacy over the integers. As shown by P. F. Stebe [39], for n ≥ 3, there exist matrices M, M′ ∈ GL(n, Z) such that (a) M and M′ are not conjugate in GL(n, Z) and (b) this fact cannot be detected by passing to modN arithmetic for some N (or by applying some other homomorphism from… view at source ↗
Figure 17
Figure 17. Figure 17: Proper cyclic orderings of the 4-cycle quiver [PITH_FULL_IMAGE:figures/full_fig_p034_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Four vortices with an apex at vertex d. Proposition 10.11. A 4-vertex quiver has a proper cyclic ordering if and only if it is not a vortex. Proof. Let Q be a 4-vertex quiver. Case 1: Q is acyclic (hence not a vortex). Then Q has a proper cyclic ordering, see Observation 10.7. Case 2: Q has an oriented 4-cycle (hence Q is not a vortex). Then the cyclic ordering induced by this cycle is proper, regardless … view at source ↗
Figure 19
Figure 19. Figure 19: Two quivers Q and µa(Q). a b c d (a, b, c, d) µa b a c d (b, a, c, d) [PITH_FULL_IMAGE:figures/full_fig_p036_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: On the left, a 4-vertex COQ Q with a proper cyclic ordering (a, b, c, d). On the right, the mutated COQ µa(Q). Its cyclic ordering (b, a, c, d) is not proper: the path c → b → d makes a left turn. Note that the quiver µa(Q) is vortex free (and possesses a totally proper cyclic ordering (a, c, b, d)). The following result appears, in different but equivalent form, in the work of D. E. Knuth [30, Section 4]… view at source ↗
Figure 21
Figure 21. Figure 21: A vortex-free quiver with no proper cyclic ordering. [PITH_FULL_IMAGE:figures/full_fig_p037_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Remark 10.19. As shown by D. E. Knuth [30, Section 6], it is NP-hard to determine whether a quiver has a vortex-free completion. a b c d e f [PITH_FULL_IMAGE:figures/full_fig_p037_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Non-vortex 4-vertex COQs, |Out(b)|≥2 (edge multiplicities not shown). Case 1: vertex b is a source. In this case, mutation does not change the cyclic ordering and µb(Q) is again proper. Case 2: we have Out(b) = {c, d}, and Q is acyclic. In this case, mutation at b reverses the arrows incident to b and leaves all other orientations unchanged. Thus µb(Q) (which has cyclic ordering (b, a, c, d)) is proper. C… view at source ↗
Figure 24
Figure 24. Figure 24: The known orientations of µb(Q), without considering the multiplicities of the arrows. Case 4: we have Out(b) ={c, d}, c→a, and Q has a 4-cycle. In this case, we know the orientations of µb(Q) shown in [PITH_FULL_IMAGE:figures/full_fig_p038_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The known orientations of µb(Q), without considering the multiplicities of the arrows [PITH_FULL_IMAGE:figures/full_fig_p038_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: For any values of a, b, c, d, e, f, g, h, i, j ≥ 4, this COQ is totally proper. Remark 12.13. The forkless part of a mutation class of quivers, introduced by M. Warkentin [40], is the set of quivers in the mutation class which are not forks. Each of the quivers discussed in Example 12.12 is the unique quiver in the forkless part of its mutation class. In order to show that a given COQ is totally proper, i… view at source ↗
Figure 27
Figure 27. Figure 27: A triangulation of a once-punctured annulus and the corresponding quiver. [PITH_FULL_IMAGE:figures/full_fig_p047_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Totally proper COQs whose quivers arise from triangulations of a 4- [PITH_FULL_IMAGE:figures/full_fig_p047_28.png] view at source ↗
read the original abstract

A cyclically ordered quiver is a quiver endowed with an additional structure of a cyclic ordering of its vertices. This structure, which naturally arises in many important applications, gives rise to new powerful mutation invariants.

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

1 major / 0 minor

Summary. The paper defines a cyclically ordered quiver as a standard quiver equipped with an additional cyclic ordering on its vertices. It asserts that this structure arises naturally in applications and produces new powerful mutation invariants.

Significance. New mutation invariants for quivers would be of interest in representation theory and cluster algebras if they are shown to be distinct from existing ones and to have concrete applications. The manuscript provides no explicit construction of any invariant, no proof of invariance under mutation, and no comparison to known invariants, so the potential significance cannot be assessed from the given text.

major comments (1)
  1. [Abstract] Abstract: the central claim that the cyclic ordering 'gives rise to new powerful mutation invariants' is stated with no derivation, no explicit formula for any invariant, no example quiver, and no verification that the claimed invariants are preserved under mutation or are independent of existing ones.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The primary concern is that the abstract's claim regarding new mutation invariants lacks supporting details in the manuscript. We address this point directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the cyclic ordering 'gives rise to new powerful mutation invariants' is stated with no derivation, no explicit formula for any invariant, no example quiver, and no verification that the claimed invariants are preserved under mutation or are independent of existing ones.

    Authors: The referee correctly observes that the provided manuscript text consists only of the definition and the unsubstantiated claim, with no explicit construction of an invariant, no formula, no example, no proof of mutation invariance, and no comparison to known invariants. This renders the significance of the claimed invariants impossible to assess from the text. We agree that these elements are required and will revise the manuscript to supply them. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract defines cyclically ordered quivers by adding a cyclic ordering to standard quivers and states that this produces new mutation invariants. No equations, derivations, predictions, or self-citations appear in the given text. No load-bearing step reduces to an input by construction, fitted parameter, or self-citation chain. The manuscript is therefore self-contained at the definitional level with no circularity to flag.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; arrays are therefore empty.

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