On cluster algebras arising from unpunctured surfaces

TL;DR

This paper provides explicit formulas for cluster variable expansions in unpunctured surface cluster algebras, proving positivity and offering new computational tools for these algebraic structures.

Contribution

It introduces a direct formula for Laurent expansions of cluster variables in unpunctured surface cluster algebras, confirming the positivity conjecture in this context.

Findings

Proved positivity conjecture for these cluster algebras.

Derived explicit Laurent polynomial formulas for cluster variables.

Provided polynomial expansion formulas in acyclic cases.

Abstract

We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the expansion of cluster variables as a polynomial whose indeterminates are the cluster variables contained in the union of an arbitrary acyclic cluster and all its neighbouring clusters in the mutation graph.