Positivity for cluster algebras from surfaces

Gregg Musiker; Ralf Schiffler; Lauren Williams

doi:10.1016/j.aim.2011.04.018

Positivity for cluster algebras from surfaces

Gregg Musiker, Ralf Schiffler, Lauren Williams

School of Mathematics

Research output: Contribution to journal › Article › peer-review

114 Scopus citations

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 language	English (US)
Pages (from-to)	2241-2308
Number of pages	68
Journal	Advances in Mathematics
Volume	227
Issue number	6
DOIs	https://doi.org/10.1016/j.aim.2011.04.018
State	Published - 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: [email protected] (G. Musiker), [email protected] (R. Schiffler), [email protected] (L. Williams).

Keywords

Cluster algebra
Positivity conjecture
Triangulated surfaces

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10.1016/j.aim.2011.04.018

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title = "Positivity for cluster algebras from surfaces",

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.",

keywords = "Cluster algebra, Positivity conjecture, Triangulated surfaces",

author = "Gregg Musiker and Ralf Schiffler and Lauren Williams",

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: [email protected] (G. Musiker), [email protected] (R. Schiffler), [email protected] (L. Williams).",

year = "2011",

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doi = "10.1016/j.aim.2011.04.018",

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AU - Musiker, Gregg

AU - Schiffler, Ralf

AU - Williams, Lauren

N1 - 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: [email protected] (G. Musiker), [email protected] (R. Schiffler), [email protected] (L. Williams).

PY - 2011/8/20

Y1 - 2011/8/20

N2 - 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.

AB - 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.

KW - Cluster algebra

KW - Positivity conjecture

KW - Triangulated surfaces

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JF - Advances in Mathematics

IS - 6

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Positivity for cluster algebras from surfaces

Abstract

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