A proof of the positivity and periodicity conjectures for T-systems

P. Di Francesco,
Institut de Physique Théorique, Saclay, France

T-systems are archetypical discrete-time integrable evolution equations,
first introduced in physics in the context of integrable quantum spin chains,
but with a wide range of mathematical connections: representation theory,
cluster algebra, combinatorics of dimers, Alternating Sign Matrices, etc.
In this talk, we present the solution of the type A T-system with various
boundaries via networks, i.e. models of weighted paths on graphs.
This explicit solution allows to show the positive Laurent phenomenon for
these systems (the solution is a Laurent polynomial with positive integer
coefficients of the initial data), and to give an alternative elementary
proof of Zamolodchikov's periodicity conjecture for the solution of the
system with wall-type boundary conditions.
(Work in collaboration with R. Kedem).