Cluster scattering diagrams and theta functions for reciprocal generalized cluster algebras

TL;DR

This paper develops a framework for constructing generalized cluster varieties, scattering diagrams, and theta functions specifically for reciprocal generalized cluster algebras, extending known structures from ordinary cluster algebras.

Contribution

It introduces new constructions of cluster varieties and scattering diagrams for reciprocal generalized cluster algebras, enabling the definition of theta functions in this context.

Findings

Constructed generalized cluster varieties and scattering diagrams for reciprocal generalized cluster algebras.

Established structural properties of theta functions in the reciprocal case.

Extended known cluster algebra structures to a broader class of generalized cluster algebras.

Abstract

We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties.