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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: A cluster expansion formula (An case). | A cluster expansion formula An case Ralf Schiffler Department of Mathematics and Statistics University of Massachusetts Amherst Amherst MA 01003-9305 USA schiffler@math.umass.edu Submitted Dec 26 2007 Accepted Apr 18 2008 Published Apr 28 2008 Mathematics Subject Classihcation 16S99 05E15 16G20 Abstract We consider the Ptolemy cluster algebras which are cluster algebras of Hnite type A with non-trivial coefficients that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed X in a Ptolemy cluster algebra we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed X. Our formula is given in a combinatorial way using paths on a triangulation of the polygon that corresponds to the seed X. 0 Introduction Cluster algebras introduced in FZ1 are commutative algebras equipped with a distinguished set of generators the cluster variables. The cluster variables are grouped into sets of constant cardinality n the clusters and the integer n is called the rank of the cluster algebra. Starting with an initial seed consisting of a cluster x together with a skew symmetrizable integer n X n matrix B bj and a coefficient vector p the set of cluster variables is obtained by repeated application of so called mutations. Ptolemy cluster algebras have been introduced by Fomin and Zelevinsky in FZ2 section 12 as examples of cluster algebras of type A. The Ptolemy algebra of rank n is described using the triangulations of a regular polygon with n 3 vertices. In this description the seeds of the cluster algebra are in bijection with the triangulations of the polygon. The cluster of the seed corresponds to the diagonals while the coefficients of the seed correspond to the boundary edges of the triangulation. The Laurent phenomenon FZ1 states that given an arbitrary seed s one can write any cluster variable of the cluster algebra as a Laurent polynomial in the cluster variables and the .
