Combinatorial interpretations for rank-two cluster algebras of affine type

Gregg Musiker, James Propp

Research output: Contribution to journalArticlepeer-review

22 Scopus citations

Abstract

Fomin and Zelevinsky [6] show that a certain two-parameter family of rational recurrence relations, here called the (b, c) family, possesses the Laurentness property: for all b, c, each term of the (b, c) sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers b, c satisfy bc < 4, the recurrence is related to the root systems of finite-dimensional rank 2 Lie algebras; when bc > 4, the recurrence is related to Kac-Moody rank 2 Lie algebras of general type [9]. Here we investigate the borderline cases bc = 4, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.

Original languageEnglish (US)
Pages (from-to)1-23
Number of pages23
JournalElectronic Journal of Combinatorics
Volume14
Issue number1 R
DOIs
StatePublished - Jan 19 2007

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