Scattering diagrams, tight gradings, and generalized positivity

• 成果类型: Article
• 署名作者: Burcroff, Amanda; Lee, Kyungyong; Mou, Lang
• 署名单位: Harvard University; University of Alabama System; University of Alabama Tuscaloosa; Korea Institute for Advanced Study (KIAS); University of Cologne
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-12229
• DOI: 10.1073/pnas.2422893122
• 发表日期: 2025-05-06
• 关键词: mirror symmetry; cluster; ALGEBRAS

摘要

In 2013, Lee, Li, and Zelevinsky introduced combinatorial objects called compatible pairs to construct the greedy bases for rank-2 cluster algebras, consisting of indecomposable positive elements including the cluster monomials. Subsequently, Rupel extended this construction to the setting of generalized rank-2 cluster algebras by defining compatible gradings. We find a class of combinatorial objects which we call tight gradings. Using this, we give a directly computable, manifestly positive, and elementary but highly nontrivial formula describing rank-2 consistent scattering diagrams. This allows us to show that the coefficients of the wall-functions on a generalized cluster scattering diagram of any rank are positive, which implies the Laurent positivity for generalized cluster algebras and the strong positivity of their theta bases.