HEXAGONbeta

Laurent and g-vector counterexamples for quivers with homotopies

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.RT — Representation Theory

Contributed by Qiyue Tang ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We give negative answers to the general Laurent-phenomenon question and the g-vector sign-coherence question for quivers with homotopies posed by Li, Liu, Mou and Pan.

Provenance statement

The counterexamples and their proofs were obtained in one round of ChatGPT-6-Pro. Codex reviewed the arguments and polished the manuscript. The results passed review by Rethlas-OSV.

References

  1. Sergey Fomin and Andrei Zelevinsky, Cluster algebras I: Foundations , Journal of the American Mathematical Society 15 (2002), 497--529. https://arxiv.org/abs/math/0104151.link
  2. Sergey Fomin and Andrei Zelevinsky, Cluster algebras IV: Coefficients , Compositio Mathematica 143 (2007), no. 1, 112--164. https://arxiv.org/abs/math/0602259.link
  3. Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations II: Applications to cluster algebras , Journal of the American Mathematical Society 23 (2010), 749--790. https://arxiv.org/abs/0904.0676.link
  4. Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich, Canonical bases for cluster algebras , Journal of the American Mathematical Society 31 (2018), 497--608. https://arxiv.org/abs/1411.1394.link
  5. Min Huang and Fang Li, Unfolding of sign-skew-symmetric cluster algebras and its applications to positivity and FF-polynomials , Advances in Mathematics 340 (2018), 221--283. https://arxiv.org/abs/1609.05981.link
  6. Fang Li and Jie Pan, Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes , arXiv:2201.01440v4, 2025. https://arxiv.org/abs/2201.01440v4.arXiv
  7. Lang Mou, Scattering diagrams for generalized cluster algebras , Algebra & Number Theory 18 (2024), 2179--2246. https://arxiv.org/abs/2110.02416.link
  8. Sergey Fomin, Michael Shapiro, and Dylan Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes , Acta Mathematica 201 (2008), 83--146. https://arxiv.org/abs/math/0608367.link
  9. Charles Paquette and Ralf Schiffler, Group actions on cluster algebras and cluster categories , Advances in Mathematics 345 (2019), 161--221. https://arxiv.org/abs/1703.06174.link
  10. Dani Kaufman, Special folding of quivers and cluster algebras , Mathematica Scandinavica 130 (2024), no. 2, 237--256. https://arxiv.org/abs/2304.07510.link
Show all 14 referencesShow fewer references
  1. Fang Li, Siyang Liu, Lang Mou, and Jie Pan, Mutations of quivers with 2-cycles , arXiv:2508.15022v3, 13 October 2026. https://arxiv.org/abs/2508.15022v3.arXiv
  2. Thomas Lam and Pavlo Pylyavskyy, Laurent phenomenon algebras , Cambridge Journal of Mathematics 4 (2016), 121--162. https://arxiv.org/abs/1206.2611.link
  3. OSVerify contributors, Rethlas-OSV , software, accessed 29 October 2026. https://github.com/OSVerify/Rethlas-OSV.link
  4. Fan Nie, Haotian Ye, Shurui Liu, Zihao Wang, Stefano Ermon, and James Zou, One-Sided Verification Under Limited Expert Review , arXiv:2610.18089v2, 2026. https://arxiv.org/abs/2610.18089v2.arXiv

Version history

  1. v1Submitted by Qiyue TangInitial depositCurrentOct 29, 2026