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T-systems with boundaries from network solutions. (English) Zbl 1266.05176
Summary: In this paper, we use the network solution of the $$A_rT$$-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 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 M. Gekhtman et al. [Electron. Res. Announc. Math. Sci. 19, 1–17 (2012; Zbl 1278.37047)].

##### MSC:
 05E10 Combinatorial aspects of representation theory 05C22 Signed and weighted graphs 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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