A Determinantal Formula for Cluster Variables in Cluster Algebras from Surfaces

Abstract

For cluster algebras of surface type, Musiker, Schiffler and Williams gave an expansion formula for cluster variables in terms of perfect matchings of snake graphs. Building on this, we provide simple determinantal formula for cluster variables via a weighting of the biadjacency matrix of the associated snake graphs, thus circumventing the enumeration of their perfect matchings.