Positivity for cluster algebras from surfaces

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.

67 pages, 45 figures, comments welcome

References in corpus (4)

• Gentle algebras arising from surface triangulations
• On cluster algebras with coefficients and 2-Calabi-Yau categories
• On cluster algebras arising from unpunctured surfaces II
• A graph theoretic expansion formula for cluster algebras of classical type