HEXAGONbeta

Strongly positive bases and quantum greedy expansions

A mix of human-written and AI-generated textHuman understanding: some partsmath.QA — Quantum Algebramath.RA — Rings and Algebrasmath.RT — Representation Theory

Contributed by 琪越 唐 ↗

Version 1 / Oct 06, 2026 / CC BY 4.0

Abstract

We prove that every element of a strongly positive basis of a coefficient-free rank-two quantum cluster algebra has nonnegative Laurent-polynomial coefficients in the quantum greedy basis. This establishes the strongly positive basis assertion of Conjecture 16 of Lee, Li, Rupel, and Zelevinsky. We also give a counterexample to their Conjecture 13(b).

Provenance statement

The results were obtained through several conversations with GPT-6-Astra and underwent repeated adversarial reviews by the same model. OpenAI Codex recon-structed and reviewed the proofs, performed exact finite algebraic checks, and prepared and revised the manuscript. 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. A. Berenstein and A. Zelevinsky, Quantum cluster algebras , Adv. Math. 195 (2005), 405--455. https://arxiv.org/abs/math/0404446 arXiv:math/0404446 .arXiv
  2. A. Berenstein and A. Zelevinsky, Triangular bases in quantum cluster algebras , Int. Math. Res. Not. IMRN (2014), no. 6, 1651--1688. https://arxiv.org/abs/1206.3586 arXiv:1206.3586 .arXiv
  3. K. Lee, L. Li, D. Rupel, and A. Zelevinsky, Greedy bases in rank 2 quantum cluster algebras , Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9712--9716. https://arxiv.org/abs/1405.2311 arXiv:1405.2311 .arXiv
  4. K. Lee, L. Li, D. Rupel, and A. Zelevinsky, The existence of greedy bases in rank 2 quantum cluster algebras , Adv. Math. 300 (2016), 360--389. https://arxiv.org/abs/1405.2414 arXiv:1405.2414 .arXiv
  5. Q. Tang, Strongly positive bases in rank-two quantum cluster algebras , Hexagon preprint 2609.00143v2 (2026). r̆l https://hexagonmath.org/2609.00143v2 .hexagon

Version history

  1. v1Submitted by 琪越 唐Initial depositCurrentOct 06, 2026