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

P. Di
Francesco,

Institut de Physique Théorique, Saclay,
France

Abstract:

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).