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Minimality of the Rank-Two Quantum Greedy Basis

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

Contributed by 琪越 唐

Version 2 / Oct 03, 2026 / CC BY 4.0

Abstract

We prove that the rank-two quantum greedy basis is the least strongly positive basis when b | c or c | b, resolving Conjecture 14 of Lee–Li–Rupel–Zelevinsky. Every strongly positive basis has nonnegative expansion coefficients in the greedy basis.

Provenance statement

The mathematical argument and manuscript were developed through several rounds of conversations with GPT-6-Astra. The manuscript subsequently underwent GPT-based review, including adversarial checks of the proof and its dependencies. Uploading this paper to this platform does not mean that I claim ownership or copyright over it. My sole intention is to let more people know that the mathematical problem addressed in this paper has been solved by AI. Once it has been reviewed and its correctness has been verified, it may be cited directly. I welcome anyone who would like to organize this work; if you do so, the credit will be yours. Email: tangqy24@mails.tsinghua.edu.cn

References

  1. K. Lee, L. Li, D. Rupel and A. Zelevinsky, Greedy bases in rank 2 quantum cluster algebras , Proc. Natl. Acad. Sci. USA 111 (2014), 9712--9716. https://pmc.ncbi.nlm.nih.gov/articles/PMC4103316/ Publisher text at PMC .link
  2. 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/html/1405.2414v1 arXiv:1405.2414v1 .arXiv
  3. A. Berenstein and A. Zelevinsky, Quantum cluster algebras , Adv. Math. 195 (2005), 405--455. https://arxiv.org/abs/math/0404446 arXiv:math/0404446 .arXiv
  4. B. Davison and T. Mandel, Strong positivity for quantum theta bases of quantum cluster algebras , Invent. Math. 226 (2021), 725--843. https://doi.org/10.1007/s00222-021-01061-1 doi:10.1007/s00222-021-01061-1 .DOI
  5. T. Mandel, Theta bases are atomic . https://arxiv.org/abs/1605.03202 arXiv:1605.03202 .arXiv
  6. Q. Tang, A Note on Universal Positivity of Rank-Two Quantum Greedy Elements , https://arxiv.org/abs/2609.37452v1 arXiv:2609.37452v1 , Proposition 5.1 and Section 6.arXiv
  7. Q. Tang, Strongly positive bases in rank-two quantum cluster algebras , https://hexagonmath.org/2609.00143v1 hexagon:2609.00143v1 (2026).hexagon

Version history

  1. v1Submitted by 琪越 唐Initial depositSupersededOct 03, 2026
  2. v2Submitted by 琪越 唐Submitted as a new versionCurrentOct 03, 2026