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 (7)

• Alternating sign matrices and domino tilings
• Quiver varieties and cluster algebras
• Cluster characters for triangulated 2-Calabi--Yau categories
• 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

Cited by in corpus (13)

• Quiver Grassmannians associated with string modules
• Quivers with potentials associated to triangulated surfaces, Part II: Arc representations
• A compendium on the cluster algebra and quiver package in sage
• Q-system Cluster Algebras, Paths and Total Positivity
• Positivity in skew-symmetric cluster algebras of finite type
• Atomic bases in cluster algebras of types A and $\tilde{A}$
• Cluster algebras of type D: pseudotriangulations approach
• Quantum F-polynomials in Classical Types
• Positivity for Regular Cluster Characters in Acyclic Cluster Algebras
• A Combinatorial Formula for Rank 2 Cluster Variables
• Combinatorial Expressions for F-polynomials in Classical Types
• A step towards the cluster positivity conjecture
• Poisson Ideals in Cluster Algebras and the Spectra of Quantized Coordinate Rings