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.
Bibliographical noteFunding Information:
✩ The first author is supported by the NSF research grant DMS-1067183; the second author is supported by the NSF research grants DMS-0908765 and DMS-1001637, and by the University of Connecticut; and the third author is supported by the NSF research grant DMS-0854432 and an Alfred Sloan Research Fellowship. * Corresponding author. E-mail addresses: email@example.com (G. Musiker), firstname.lastname@example.org (R. Schiffler), email@example.com (L. Williams).
- Cluster algebra
- Positivity conjecture
- Triangulated surfaces