Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài:Positivity in coefficient-free rank two cluster algebras. | Positivity in coefficient-free rank two cluster algebras G. Dupont Universite de Lyon Universite Lyon 1 Institut Camille Jordan 43 Bd du 11 novembre 1918 F-69622 Villeurbanne cedex dupont@ Submitted Jan 30 2009 Accepted Jul 30 2009 Published Aug 7 2009 Mathematics Subject Classification 13F60 16G20 05E10 05E40 Abstract Let b c be positive integers x1 x2 be indeterminates over Z and xm m E Z be rational functions defined by xm-1xm 1 Xm 1 if m is odd and xm-1xm 1 xt 1 if m is even. In this short note we prove that for any m k E Z xk can be expressed as a substraction-free Laurent polynomial in Z xm1 x m 1 . This proves Fomin-Zelevinsky s positivity conjecture for coefficient-free rank two cluster algebras. Introduction A combinatorial result Let b c be positive integers and x1 x2 be indeterminates over Z. The coefficient-free cluster algebra A b c is the subring of the field Q x1 x2 generated by the elements xm m E Z satisfying the recurrence relations 1 x if m E 2Z 1 xm-1 Xm 1 - 1 if m E 2Z. 1 xm-1 The elements xm m E Z are called the cluster variables of A b c and the pairs xm xm 1 m E Z are called the clusters of A b c . The Laurent phenomenon FZ02 implies that for any m E Z and any k E Z the cluster variable xk belongs to the ring of Laurent polynomials Z rntxt J- When bc 4 it was proved by Sherman-Zelevinsky SZ04 and independently by Musiker-Propp MP06 that for any m E Z and any k E Z the cluster variable xk belongs the electronic journal of combinatorics 16 2009 R98 1 to N ximu This was later proved by Caldero-Reineke for any b c CR08 . In this paper we prove this for arbitrary positive integers b c. More precisely the main result of the paper is Theorem 8. Let b c be positive integers. With the above notations we have Xk k G Z c N xS-i for any m G Z. The positivity conjecture for cluster algebras In particular this result is a particular case of a general conjecture formulated by Fomin and Zelevinsky for arbitrary cluster algebras. We recall
