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Báo cáo toán học: "he alternating sign matrix polytope"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The alternating sign matrix polytope. | The alternating sign matrix polytope Jessica Striker School of Mathematics University of Minnesota Minneapolis MN 55455 jessica@math.umn.edu Submitted Jan 4 2008 Accepted Mar 16 2009 Published Mar 25 2009 Mathematics Subject Classification 05C50 52B05 Abstract We define the alternating sign matrix polytope as the convex hull of n X n alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove that the dimension of any face can be easily determined from this characterization. 1 Introduction and background The Birkhoff polytope which we will denote as Bn has been extensively studied and generalized. It is defined as the convex hull of the n X n permutation matrices as vectors in Rn . Many analogous polytopes have been studied which are subsets of Bn see e.g. 8 . In contrast we study a polytope containing Bn. We begin with the following definitions. Definition 1.1. Alternating sign matrices ASMs are square matrices with the following properties entries G 0 1 -1 the entries in each row and column sum to 1 nonzero entries in each row and column alternate in sign THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R41 1 100 100 010 0 1 0 010 001 100 1 -1 1 001 010 001 0 10 0 1 0 w 0 0 1 w 0 0 1 001 100 010 100 010 100 Figure 1 The 3 X 3 ASMs The total number of n X n alternating sign matrices is given by the expression TT 3j 1 j 0 n j 1 Mills Robbins and Rumsey conjectured this formula 16 and then over a decade later Doron Zeilberger proved it 21 . Shortly thereafter Kuperberg gave a shorter proof using a bijection .