Scattering diagrams for generalized cluster algebras

TL;DR

This paper extends the concept of scattering diagrams to generalized cluster algebras with binomial exchange polynomials, revealing their structure and positivity properties, and connecting them to theta functions.

Contribution

It constructs scattering diagrams for generalized cluster algebras, generalizing previous diagrams and demonstrating their relation to theta functions and positivity.

Findings

Generalized cluster variables are theta functions.

Scattering diagrams exhibit positivity and cluster complex structures.

The work generalizes cluster scattering diagrams to a broader class of algebras.

Abstract

We construct scattering diagrams for Chekhov-Shapiro's 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.