Cluster scattering diagrams via quiver moduli and tight gradings

TL;DR

This paper investigates rank-2 cluster scattering diagrams using quiver moduli and tight gradings, proving conjectures about their structure and symmetries with new combinatorial and quiver-theoretic methods.

Contribution

It introduces a novel approach combining quiver theory and tight gradings to analyze scattering diagrams, extending previous conjectures and providing new proofs of symmetry properties.

Findings

Proved and extended conjectures on wall-function coefficients.

Established a new proof of Weyl group symmetry.

Connected quiver moduli with cluster scattering diagram properties.

Abstract

We study rank-2 cluster scattering diagrams through moduli spaces of quiver representations and a recently developed combinatorial framework of tight gradings. Combining quiver-theoretic and combinatorial methods, we prove and extend a collection of conjectures posed by Elgin--Reading--Stella concerning the structural and enumerative properties of the wall-function coefficients. The tight grading perspective also provides a new proof of the Weyl group symmetry of the scattering diagram.