tailieunhanh - Báo cáo toán học: "The solution of the Ar T-system for arbitrary boundary"

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: The solution of the Ar T-system for arbitrary boundary. | The solution of the Ar T-system for arbitrary boundary Philippe Di Francesco Department of Mathematics University of Michigan 530 Church Street Ann Arbor MI 48190 USA and Institut de Physique Theorique du Commissariat a l Energie Atomique Unite de recherche associeee du CNRS CEA Saclay IPhT Bat 774 F-91191 Gif sur Yvette Cedex France Submitted Feb 23 2010 Accepted Jun 3 2010 Published Jun 14 2010 Mathematics Subject Classification 05C88 Abstract We present an explicit solution of the Ar T-system for arbitrary boundary conditions. For each boundary this is done by constructing a network . a graph with positively weighted edges and the solution is expressed as the partition function for a family of non-intersecting paths on the network. This proves in particular the positive Laurent property namely that the solutions are all Laurent polynomials of the initial data with non-negative integer coefficients. 1 Introduction In this paper we study the solutions of the Ar T-system namely the following coupled system of recursion relations for a j k G Z Tj iTj-l T i T ij t T -i k for a G Ir 1 2 . r and subject to the boundary conditions T0 j fc Tr i j fc 1 j k G Z This system arose in many different contexts. The system and its generalizations were introduced as the set of relations satisfied by the eigenvalues of the fused transfer matrices of generalized quantum spin chains based on any simply-laced Lie algebra g 2 16 in this paper we restrict ourselves to the case g slr 1 but we believe our constructions can be adapted to other g s as well. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R89 1 With the additional condition that Ta 0 k 1 k E Z and the restriction to j E Z the solutions of were also interpreted as the q-characters of some representations of the affine Lie algebra Uq slr 1 the so-called Kirillov-Reshetikhin modules 12 indexed by a E Ir 1 2 . r and j E Z while k stands for a discrete spectral .