On cluster algebras arising from unpunctured surfaces II

By Ralf Schiffler. 2008

70 citations recorded by the source corpus.

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. 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. Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincare characteristic of quiver Grassmannians in Dynkin type $A$ and affine Dynkin type $\tilde A$.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a42c0e59be7dfc03142a68fac14d91c3d279fb68

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Citations in the corpus

Cluster expansion formulas and perfect matchings (2008, methodology): “In [ 32 ], building on earlier results in [31, 33], this model was used to give a direct expansion formula for cluster variables in cluster algebras associated to unpunctured surfaces, with arbitrary coefficients, in terms of certain paths on the triangulation.”
Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs (2014, result): “In [23], building on earlier work [22,27–29], the authors gave a combinatorial formula for the cluster variables in cluster algebras of surface type.”