T-systems with boundaries from network solutions

2012/08/21 by Di Francesco, Philippe, Kedem, Rinat
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1208.4333

Abstract

In this paper, we use the network solution of the A_r T-system to derive that of the unrestricted A_∞ T-system, equivalent to the octahedron relation. We then present a method for implementing various boundary conditions on this system, which consists of picking initial data with suitable symmetries. The corresponding restricted T-systems are solved exactly in terms of networks. This gives a simple explanation for phenomena such as the Zamolodchikov periodicity property for T-systems (corresponding to the case A_ℓ× A_r) and a combinatorial interpretation for the positive Laurent property of the variables of the associated cluster algebra. We also explain the relation between the T-system wrapped on a torus and the higher pentagram maps of Gekhtman et al.

Related

• Positivity of the T-system cluster algebra
• Difference equations arising from cluster algebras
• Dimers, networks, and cluster integrable systems
• Integrable systems and cluster algebras
• Arctic curves of the T-system with Slanted Initial Data