Cluster expansion formulas and perfect matchings

Gregg Musiker; Ralf Schiffler

doi:10.1007/s10801-009-0210-3

Cluster expansion formulas and perfect matchings

Gregg Musiker, Ralf Schiffler

School of Mathematics

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

50 Scopus citations

Abstract

We study cluster algebras with principal 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 perfect matchings of a certain graph G_T,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph G_T,γ..

Original language	English (US)
Pages (from-to)	187-209
Number of pages	23
Journal	Journal of Algebraic Combinatorics
Volume	32
Issue number	2
DOIs	https://doi.org/10.1007/s10801-009-0210-3
State	Published - Sep 2010

Bibliographical note

Funding Information:
The first author is supported by an NSF Mathematics Postdoctoral Fellowship, and the second author is supported by the NSF grant DMS-0908765 and by the University of Connecticut.

Keywords

Cluster algebra
F-polynomial
Principal coefficients
Snake graph
Triangulated surface

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10.1007/s10801-009-0210-3

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abstract = "We study cluster algebras with principal 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 perfect matchings of a certain graph GT,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph GT,γ..",

keywords = "Cluster algebra, F-polynomial, Principal coefficients, Snake graph, Triangulated surface",

author = "Gregg Musiker and Ralf Schiffler",

note = "Funding Information: The first author is supported by an NSF Mathematics Postdoctoral Fellowship, and the second author is supported by the NSF grant DMS-0908765 and by the University of Connecticut.",

year = "2010",

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doi = "10.1007/s10801-009-0210-3",

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AU - Musiker, Gregg

AU - Schiffler, Ralf

N1 - Funding Information: The first author is supported by an NSF Mathematics Postdoctoral Fellowship, and the second author is supported by the NSF grant DMS-0908765 and by the University of Connecticut.

PY - 2010/9

Y1 - 2010/9

N2 - We study cluster algebras with principal 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 perfect matchings of a certain graph GT,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph GT,γ..

AB - We study cluster algebras with principal 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 perfect matchings of a certain graph GT,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph GT,γ..

KW - Cluster algebra

KW - F-polynomial

KW - Principal coefficients

KW - Snake graph

KW - Triangulated surface

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ER -

Cluster expansion formulas and perfect matchings

Abstract

Bibliographical note

Keywords

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