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arXiv · 0809.2593

On cluster algebras arising from unpunctured surfaces II

Ralf Schiffler
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Abstract

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-Poincaré characteristic of quiver Grassmannians in Dynkin type $A$ and affine Dynkin type $\tilde A$.

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Ralf Schiffler. 2008-09-18. On cluster algebras arising from unpunctured surfaces II. https://arxiv.org/abs/0809.2593

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