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The support conjecture for rank-two quantum triangular bases

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 2 / Sep 30, 2026 / CC BY 4.0

Abstract

We prove Conjecture 11 of Lee, Li, Rupel, and Zelevinsky on the support of triangular basis elements in rank-two quantum cluster algebras.

Provenance statement

The main results of this paper were generated using GPT-6-Astra and verified by the Danus system. 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. Additionally, if you need more related literature or detailed materials, you can contact me by email. Email: tangqy24@mails.tsinghua.edu.cn

References

  1. A. Berenstein and A. Zelevinsky, Triangular bases in quantum cluster algebras , Int. Math. Res. Not. IMRN 2014 (2014), no. 6, 1651--1688. https://arxiv.org/abs/1206.3586 arXiv:1206.3586 ; version 2.arXiv
  2. M. A. de Cataldo and L. Migliorini, The decomposition theorem, perverse sheaves and the topology of algebraic maps , Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 4, 535--633. https://arxiv.org/abs/0712.0349v2 arXiv:0712.0349 ; version 2.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 ; version 1.arXiv
  4. L. Li, Nakajima's quiver varieties and triangular bases of rank-2 cluster algebras , J. Algebra 634 (2023), 97--164. https://arxiv.org/abs/2208.12307 arXiv:2208.12307 ; version 2.arXiv
  5. L. Li, Nakajima's quiver varieties and triangular bases of bipartite cluster algebras , in Representation Theory and Flag Varieties , Contemp. Math. 837 (2026), 131--158. https://arxiv.org/abs/2405.08234 arXiv:2405.08234 . The result numbering used here refers to version 1 of the preprint.arXiv

Version history

  1. v1Submitted by 琪越 唐Initial depositSupersededSep 30, 2026
  2. v2Submitted by 琪越 唐Submitted as a new versionCurrentOct 04, 2026