Positivity for cluster algebras from surfaces

TL;DR

This paper provides combinatorial formulas for Laurent expansions of cluster variables in surface-based cluster algebras, proving the positivity conjecture in geometric type for these cases.

Contribution

It introduces explicit combinatorial formulas for Laurent expansions in cluster algebras from surfaces, confirming the positivity conjecture in this setting.

Findings

Proved positivity conjecture for cluster algebras from surfaces

Derived combinatorial formulas for Laurent expansions

Applicable to surfaces with or without punctures and arbitrary seeds

Abstract

We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.