A determinantal formula for cluster variables in cluster algebras from surfaces

TL;DR

This paper introduces a straightforward determinantal formula for cluster variables in surface-type cluster algebras, simplifying previous combinatorial methods by avoiding perfect matchings enumeration.

Contribution

It presents a novel determinantal approach using the weighted biadjacency matrix, streamlining the computation of cluster variables in surface cluster algebras.

Findings

Determinantal formula simplifies computation of cluster variables

Circumvents complex perfect matchings enumeration

Provides a new algebraic tool for surface cluster algebras

Abstract

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