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 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.
@article{10_37236_2645,
author = {Philippe Di Francesco and Rinat Kedem},
title = {T-systems with boundaries from network solutions},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {1},
doi = {10.37236/2645},
zbl = {1266.05176},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2645/}
}
TY - JOUR
AU - Philippe Di Francesco
AU - Rinat Kedem
TI - T-systems with boundaries from network solutions
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/2645/
DO - 10.37236/2645
ID - 10_37236_2645
ER -
%0 Journal Article
%A Philippe Di Francesco
%A Rinat Kedem
%T T-systems with boundaries from network solutions
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/2645/
%R 10.37236/2645
%F 10_37236_2645
Philippe Di Francesco; Rinat Kedem. T-systems with boundaries from network solutions. The electronic journal of combinatorics, Tome 20 (2013) no. 1. doi: 10.37236/2645
