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Báo cáo toán học: "Positivity of the T-system cluster algebra"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Positivity of the T-system cluster algebra. | Positivity of the T-system cluster algebra Philippe Di Francesco Insitut de Physique Theorique du Commissariat à l Energie Atomique Unite de recherche associeee du CNRS CEA Saclay IPhT Bat 774 F-91191 Gif sur Yvette Cedex France philippe.di-francesco@cea.fr Rinat Kedem Department of Mathematics University of Illinois Urbana IL 61821 USA rinat@illinois.edu Submitted Sep 10 2009 Accepted Nov 12 2009 Published Nov 24 2009 Mathematics Subject Classification 05C88 Abstract We give the path model solution for the cluster algebra variables of the T-system of type Ar with generic boundary conditions. The solutions are partition functions of strongly non-intersecting paths on weighted graphs. The graphs are the same as those constructed for the Q-system in our earlier work and depend on the seed or initial data in terms of which the solutions are given. The weights are time-dependent where time is the extra parameter which distinguishes the T-system from the Q-system usually identified as the spectral parameter in the context of representation theory. The path model is alternatively described on a graph with non-commutative weights and cluster mutations are interpreted as non-commutative continued fraction rearrangements. As a consequence the solution is a positive Laurent polynomial of the seed data. 1 Introduction In this paper we study solutions of the T-system associated to the Lie algebras Ar which we write in the following form To j k iTa j k-i T J ỈJ1 T t-I k Ta ujT where j k G Z a G Ir 1 . r and with boundary conditions To j k Tr I j fc 1 j k G Z. 1.1 1.2 THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R140 1 We consider these equations to be discrete evolution equations for the commutative variables Ta jk in the direction of the discrete variable k. Originally this relation appeared as the fusion relation for the commuting transfer matrices of the generalized Heisenberg model 1 18 associated with a simply-laced Lie algebra g where it is written in the form Taj.k .