Positivity for cluster algebras from surfaces. (English) Zbl 1331.13017
Summary: 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 S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] for cluster algebras from surfaces, in geometric type.
MSC:
| 13F60 | Cluster algebras |
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
Citations:
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