Cluster Algebras and Scattering Diagrams, Part III. Cluster Scattering Diagrams

TL;DR

This paper provides a comprehensive, self-contained study of cluster scattering diagrams, focusing on their construction, mutation invariance, and positivity of theta functions, emphasizing the roles of dilogarithm elements and the pentagon relation.

Contribution

It offers detailed proofs and clarifies fundamental properties of cluster scattering diagrams, enhancing understanding of their structure and invariance under mutation.

Findings

Construction and mutation invariance of cluster scattering diagrams proved

Positivity of theta functions established

Role of dilogarithm elements and pentagon relation highlighted

Abstract

This is a self-contained exposition of several fundamental properties of cluster scattering diagrams introduced and studied by Gross, Hacking, Keel, and Kontsevich. In particular, detailed proofs are presented for the construction, the mutation invariance, and the positivity of theta functions of cluster scattering diagrams. Throughout the text we highlight the fundamental roles of the dilogarithm elements and the pentagon relation in cluster scattering diagrams.