tailieunhanh - Báo cáo toán học: "Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type."

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type. | Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type Gregg Musiker Department of Mathematics University of California San Diego La Jolla CA 92093-0112 gmusiker@ James Propp Department of Mathematical Sciences University of Massachusetts Lowell Lowell MA 01854 propp@ Submitted Feb 20 2006 Accepted Jan 11 2007 Published Jan 19 2007 Mathematics Subject Classification 05A99 05C70 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. 1 Introduction In 5 6 Fomin and Zelevinsky prove that for all positive integers b and c the sequence of rational functions xn n 0 satisfying the b c -recurrence i xn _i 1 xn-2 for n odd t x _1 1 xn_2 for n even THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R15 1 is a sequence of Laurent polynomial in the variables x1 and x2 that is for all n 2 xn can be written as a sum of Laurent monomials of the form axi1 x2 where the coefficient a is an integer and i and j are not necessarily positive integers. In fact Fomin and Zelevinsky conjecture that the coefficients are always positive integers. It is worth mentioning that variants of .