Cluster scattering diagrams of acyclic affine type

TL;DR

This paper constructs explicit cluster scattering diagrams for acyclic affine type exchange matrices, showing their equivalence with mutation fans and fans from the almost-positive roots model, advancing understanding of cluster algebra structures.

Contribution

It provides an explicit construction of cluster scattering diagrams for acyclic affine types and establishes their equivalence with known fans, clarifying their geometric and combinatorial properties.

Findings

Cluster scattering fan coincides with the mutation fan.

Cluster scattering fan matches the fan from the almost-positive roots model.

Explicit construction aids in understanding affine type cluster algebras.

Abstract

We give an explicit construction of the cluster scattering diagram for any acyclic exchange matrix of affine type. We show that the corresponding cluster scattering fan coincides both with the mutation fan and with a fan constructed in the almost-positive roots model.