HEXAGONbeta

An explicit nonideal rooted cluster morphism

A mix of human-written and AI-generated textHuman understanding: some partsmath.RA — Rings and Algebrasmath.CO — Combinatorics

Contributed by Qiyue Tang

Version 2 / Oct 05, 2026 / CC BY 4.0

Abstract

We give a counterexample to Chang and Zhu’s question of whether every explicit rooted cluster morphism is ideal.

Provenance statement

While surveying open questions in Bin Zhu’s papers on cluster algebras, I encountered Problem 2 in his joint paper with Wen Chang. Through two or three rounds of discussion with GPT-6-Astra, a counterexample and its proof were developed. The argument then underwent a separate, strongly adversarial review using GPT-6-Astra. Uploading this paper to this platform does not mean that I claim ownership or copyright over it. My sole intention is to share this AI-generated solution and make it available for independent review. Once its correctness has been independently verified, the paper may be cited directly. I welcome anyone who would like to organize this work; if you do so, the credit will be yours. I also welcome further research building on this work, including extensions, new applications, and related questions. For additional references or detailed research materials, please contact me by email. Email: tangqy24@mails.tsinghua.edu.cn

References

  1. I. Assem, G. Dupont and R. Schiffler, On a category of cluster algebras , J. Pure Appl. Algebra 218 (2014), 553--582. arXiv:1201.5986v1 (https://arxiv.org/abs/1201.5986v1).arXiv
  2. W. Chang and B. Zhu, On rooted cluster morphisms and cluster structures in 2-Calabi--Yau triangulated categories , J. Algebra 458 (2016), 387--421. arXiv:1410.5702v2 (https://arxiv.org/abs/1410.5702v2).arXiv
  3. S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations , J. Amer. Math. Soc. 15 (2002), 497--529. arXiv:math/0104151 (https://arxiv.org/abs/math/0104151).arXiv
  4. S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification , Invent. Math. 154 (2003), 63--121. arXiv:math/0208229v2 (https://arxiv.org/abs/math/0208229v2).arXiv
  5. M. Huang, F. Li and Y. Yang, On structure of cluster algebras of geometric type I: In view of sub-seeds and seed homomorphisms , Sci. China Math. 61 (2018), 831--854. arXiv:1509.01050v4 (https://arxiv.org/abs/1509.01050v4). The question cited here follows Theorem 6.7 in this arXiv version.arXiv
  6. B. Keller, P.-G. Plamondon and F. Qin, A refined multiplication formula for cluster characters , arXiv:2301.01059v2 (https://arxiv.org/abs/2301.01059v2).arXiv
  7. G. Muller, Skein and cluster algebras of marked surfaces , Quantum Topol. 7 (2016), 435--503. doi:10.4171/QT/79 (https://doi.org/10.4171/QT/79).DOI

Version history

  1. v1 · 8 pagesSubmitted by 琪越 唐Initial depositSupersededOct 05, 2026
  2. v2 · 8 pagesSubmitted by Qiyue TangSubmitted as a new versionCurrentNov 06, 2026