tailieunhanh - Báo cáo toán học: "Weighted Aztec Diamond Graphs and the Weyl Character Formula"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:Weighted Aztec Diamond Graphs and the Weyl Character Formula. | Weighted Aztec Diamond Graphs and the Weyl Character Formula Georgia Benkart Department of Mathematics University of Wisconsin Madison WI 53706 e-mail benkart@ Oliver Eng Epic Systems Corporation 5301 Tokay Blvd. Madison WI 53711 e-mail oeng@ Submitted Nov 19 2002 Accepted Jan 20 2004 Published Apr 2 2004 MR Subject Classification 52C20 05B45 17B10 Keywords Aztec diamonds domino tilings Weyl character formula Abstract Special weight labelings on Aztec diamond graphs lead to sum-product identities through a recursive formula of Kuo. The weight assigned to each perfect matching of the graph is a Laurent monomial and the identities in these monomials combine to give Weyl s character formula for the representation with highest weight p the half sum of the positive roots for the classical Lie algebras. Choose a positive integer n and label the 2n X 2n checkerboard matrix style. The Aztec diamond of order n is the subset of this checkerboard consisting of the squares whose coordinates i j satisfy j i n and n 1 i j 3n 1 . Thus in an Aztec diamond of order n there will be 2n rows having 2 4 . . . 2n 2n . 4 2 squares from top to bottom as in Figure 1. A domino covers two adjacent squares and the number of domino tilings of the Aztec diamond of order n is 2n n 1 2 by EKLP1 EKLP2 . Those Support from National Science Foundation grant DMS-9970119 is gratefully acknowledged. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R28 1 papers establish connections between domino tilings of Aztec diamonds and alternating sign matrices which in turn are related to a host of topics such as states in the square ice model complete monotone triangles and descending plane partitions see for example Br . A monotone triangle is a triangular array T of positive integers which strictly increase from left to right along its rows and weakly increase left to right along all of its diagonals. When the bottom row consists of 1 2 . as in the example below then T is said to .