tailieunhanh - Báo cáo tin học: "Higher Spin Alternating Sign Matrices"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Higher Spin Alternating Sign Matrices. | Higher Spin Alternating Sign Matrices Roger E. Behrend and Vincent A. Knight School of Mathematics Cardiff University Cardiff CF24 4AG UK behrendr@ knightva@ Submitted Aug 28 2007 Accepted Nov 25 2007 Published Nov 30 2007 Mathematics Subject Classifications 05A15 05B20 52B05 52B11 82B20 82B23 Abstract We define a higher spin alternating sign matrix to be an integer-entry square matrix in which for a nonnegative integer r all complete row and column sums are r and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r 2 statistical mechanical vertex models with domain-wall boundary conditions. The case r 1 gives standard alternating sign matrices while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension n 1 2 whose vertices are the standard alternating sign matrices of size n. It then follows that for fixed n these matrices are enumerated by an Ehrhart polynomial in r. Keywords alternating sign matrix semimagic square convex polytope higher spin vertex model THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R83 1 1. Introduction Alternating sign matrices are mathematical objects with intriguing combinatorial properties and important connections to mathematical physics and the primary aim of this paper is to introduce natural generalizations of these matrices which also seem to display interesting such properties and connections. Alternating sign matrices were first defined in 50 and the significance of their connection with mathematical physics first became apparent in 47 in which a determinant formula for the partition function of an integrable statistical mechanical model and a simple correspondence between configurations of that model and alternating sign matrices were used