Positivity for cluster algebras from surfaces

Gregg Musiker, Ralf Schiffler, Lauren Williams

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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.

Original languageEnglish (US)
Pages (from-to)2241-2308
Number of pages68
JournalAdvances in Mathematics
Volume227
Issue number6
DOIs
StatePublished - Aug 20 2011

Bibliographical note

Funding 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: musiker@math.umn.edu (G. Musiker), schiffler@math.uconn.edu (R. Schiffler), williams@math.berkeley.edu (L. Williams).

Keywords

  • Cluster algebra
  • Positivity conjecture
  • Triangulated surfaces

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