T-systems with boundaries from network solutions

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 for 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.