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Báo cáo toán học: "Counting the number of elements in the ˜ mutation classes of An −quiver"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Counting the number of elements in the ˜ mutation classes of An −quivers. | Counting the number of elements in the mutation classes of An-quivers Janine Bastian Institut fur Algebra Zahlentheorie und Diskrete Mathematik Leibniz Universitat Hannover Welfengarten 1 D-30167 Hannover Germany bastian@math.uni-hannover.de Thomas Prellberg School of Mathematical Sciences Queen Mary University of London Mile End Road London E1 4NS United Kingdom t.prellberg@qmul.ac.uk Martin Rubey Institut fuur Algebra Zahlentheorie und Diskrete Mathematik Leibniz Universituat Hannover Welfengarten 1 D-30167 Hannover Germany mart in.rubey@math.uni-hannover.de Christian Stump Laboratoire de combinatoire et d informatique mathematique Universite du Quebec a Montreal President-Kennedy Montreal Quebec H2X 3Y7 Canada christian.stump@lacim.ca Submitted Oct 17 2010 Accepted Apr 20 2011 Published Apr 29 2011 Mathematics Subject Classification 05A15 16G20 Abstract In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type An in the derived equivalence classes. In particular we obtain the number of elements in the mutation classes of quivers of type An. As a by-product this provides an alternative proof for the number of quivers mutation equivalent to a quiver of Dynkin type Dn which was first determined by Buan and Torkildsen in 5 . THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R00 1 1 Introduction Quiver mutation is a central element in the recent theory of cluster algebras introduced by Fomin and Zelevinsky in 10 . It is an elementary operation on quivers which generates an equivalence relation. The mutation class of a quiver Q is the class of all quivers which are mutation equivalent to Q. The mutation class of quivers of type An is the class containing all quivers mutation equivalent to a quiver whose underlying graph is the Dynkin diagram of type An shown in Figure 1 a . This mutation class was described by Caldero Chapoton and Schiffler 7 in terms of triangulations. An explicit characterisation of the quivers .