Combinatorial cluster expansion formulas from triangulated surfaces

TL;DR

This paper introduces a simplified combinatorial formula for cluster expansions in cluster algebras from triangulated surfaces, connecting perfect matchings of angles, snake graphs, bipartite graphs, and quivers with potential.

Contribution

It provides a new, more straightforward combinatorial formula for cluster expansions, establishing bijections among various perfect matchings and cuts.

Findings

Simplified cluster expansion formula for triangulated surfaces.

Established bijections between perfect matchings of angles, snake graphs, and bipartite graphs.

Unified combinatorial framework connecting different representations in cluster algebras.

Abstract

We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles. Our formula simplifies the cluster expansion formula given by Musiker-Schiffler-Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between perfect matchings of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential.