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Scattering diagrams for generalized cluster algebras

Lang Mou

Vol. 18 (2024), No. 12, 2179–2246
Abstract

We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors.

Keywords
generalized cluster algebra, scattering diagram, cluster variety
Mathematical Subject Classification
Primary: 13F60
Milestones
Received: 28 November 2022
Revised: 31 October 2023
Accepted: 22 January 2024
Published: 21 October 2024
Authors
Lang Mou
Mathematisches Institut
Universität zu Köln
Köln
Germany

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