The solution of the $A_r$ T-system for arbitrary boundary

Abstract

We present an explicit solution of the $A_r$ $T$-system for arbitrary boundary conditions. For each boundary, this is done by constructing a network, i.e. 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.

42 pages, 33 figures

References in corpus (5)

• Total positivity, Grassmannians, and networks
• Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property
• Perfect Matchings and the Octahedron Recurrence
• Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich
• A Periodicity Theorem for the Octahedron Recurrence

Cited by in corpus (1)

• Noncommutative integrability, paths and quasi-determinants