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Báo cáo toán học: "Semicanonical basis generators of the cluster algebra (1) of type A1"
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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Semicanonical basis generators of the cluster algebra (1) of type A1. | Semicanonical basis generators of the cluster algebra of type A11 Andrei Zelevinsky Department of Mathematics Northeastern University Boston USA andrei@neu.edu Submitted Jul 27 2006 Accepted Dec 23 2006 Published Jan 19 2007 Mathematics Subject Classification 16S99 Abstract We study the cluster variables and imaginary elements of the semicanonical basis for the coefficient-free cluster algebra of affine type A11 . A closed formula for the Laurent expansions of these elements was given by P.Caldero and the author. As a by-product there was given a combinatorial interpretation of the Laurent polynomials in question equivalent to the one obtained by G.Musiker and J.Propp. The original argument by P.Caldero and the author used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton. This note provides a quick self-contained and completely elementary alternative proof of the same results. 1 Introduction The coefficient-free cluster algebra A of type A11 is a subring of the field Q x1 X2 generated by the elements xm for m 2 Z satisfying the recurrence relations Xm 1 Xm 1 X2m 1 m 2 Z . 1 This is the simplest cluster algebra of infinite type it was studied in detail in 2 6 . Besides the generators xm called cluster variables A contains another important family of elements s0 s1 . defined recursively by So 1 S1 XoX3 - X1X2 Sn S1 Sn-1 - Sn-2 n 2 . 2 Research supported by NSF DMS grant 0500534 and by a Humboldt Research Award THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N4 1 As shown in 2 6 the elements s1 s2 together with the cluster monomials xmx i for all m 2 Z and P q 0 form a Z-basis of A referred to as the semicanonical basis. As a special case of the Laurent phenomenon established in 3 A is contained in the Laurent polynomial ring Z xịt1 x 1 . In particular all xm and sn can be expressed as integer Laurent polynomials in x1 and x2. These Laurent polynomials were explicitly computed in 2 using their geometric interpretation due to P. .