T-systems with boundaries from network solutions

TL;DR

This paper uses network solutions to analyze boundary conditions in T-systems, providing exact solutions, explaining periodicity phenomena, and connecting to cluster algebras and higher pentagram maps.

Contribution

It introduces a method for implementing boundary conditions in T-systems via network solutions, offering exact solutions and new insights into their properties.

Findings

Exact solutions for restricted T-systems using networks

Explanation of Zamolodchikov periodicity in T-systems

Connection between T-systems on a torus and higher pentagram maps

Abstract

In this paper, we use the network solution of the $A_r$ $T$-system to derive that of the unrestricted $A_\infty$ $T$-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 $A_\ell\times A_r$) 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.