TAILIEUCHUNG - Báo cáo toán học: " New Graphs of Finite Mutation Type"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: New Graphs of Finite Mutation Type. | New Graphs of Finite Mutation Type Harm Derksen Department of Mathematics University of Michigan hderksen@ Theodore Owen Department of Mathematics Iowa State University owentheo@ Submitted Apr 21 2008 Accepted Nov 3 2008 Published Nov 14 2008 Mathematics Subject Classification 05E99 Abstract To a directed graph without loops or 2-cycles we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices and more generally skew-symmetrizable matrices have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph r is the set of all isomorphism classes of graphs that can be obtained from r by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with marked points. We will call such graphs of geometric type . Besides graphs with 2 vertices and graphs of geometric type there are only 9 other exceptional mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes. Cluster algebras were introduced by Fomin and Zelevinsky in 5 6 to create an algebraic framework for total positivity and canonical bases in semisimple algebraic groups. An n X n matrix B bi j is called skew symmetrizable if there exists nonzero d1 d2 . dn such that dibij djbj i for all i j. An exchange matrix is a skew-symmetrizable matrix with integer entries. A seed is a pair x B where B is an exchange matrix and x x1 x2 . xng is a set of n algebraically independent elements. For any k with 1 k n we define another This first author is partially supported by NSF grant DMS 0349019. This grant also supported the REU research project of the second author on which this paper is based. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R139 1 seed x0 B0 k x B as follows. The matrix B0 bij is given by b0 i bi j if i k

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