A combinatorial approach to scattering diagrams

TL;DR

This paper explores the structure of cluster scattering diagrams, computes specific functions in affine types, and constructs diagrams from Cambrian fans, advancing understanding in mirror symmetry and cluster algebras.

Contribution

It provides explicit calculations for limiting walls in rank-2 affine types and constructs diagrams from Cambrian fans, linking scattering diagrams to combinatorial objects.

Findings

Recovered Reineke's formula for rank-2 affine types

Identified Narayana numbers in the context of cluster variables

Constructed diagrams from Cambrian fans and sortable elements

Abstract

Scattering diagrams arose in the context of mirror symmetry, but a special class of scattering diagrams (the cluster scattering diagrams) were recently developed to prove key structural results on cluster algebras. We use the connection to cluster algebras to calculate the function attached to the limiting wall of a rank-2 cluster scattering diagram of affine type. In the skew-symmetric rank-2 affine case, this recovers a formula due to Reineke. In the same case, we show that the generating function for signed Narayana numbers appears in a role analogous to a cluster variable. In acyclic finite type, we construct cluster scattering diagrams of acyclic finite type from Cambrian fans and sortable elements, with a simple direct proof.