Cluster expansion formulas and perfect matchings

TL;DR

This paper provides explicit formulas for cluster variable expansions in cluster algebras from surfaces, using perfect matchings and subgraphs of a constructed graph, enhancing combinatorial understanding.

Contribution

It introduces direct combinatorial formulas for Laurent expansions of cluster variables in surface-associated cluster algebras, linking algebraic and graph-theoretic methods.

Findings

Explicit formula using perfect matchings of a constructed graph.

Alternative formula based on subgraphs of the same graph.

Application to cluster algebras from unpunctured surfaces.

Abstract

We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$.