Positivity of the T-System Cluster Algebra

We give the path model solution for the cluster algebra variables of the T system of type Ar with generic boundary conditions. The solutions are partition functions of (strongly) non-intersecting paths on weighted graphs. The graphs are the same as those constructed for the Q-system in our earlier work, and depend on the seed or initial data in terms of which the solutions are given. The weights are “time-dependent” where “time” is the extra parameter which distinguishes the T -system from the Q-system, usually identified as the spectral parameter in the context of representation theory. The path model is alternatively described on a graph with non-commutative weights, and cluster mutations are interpreted as noncommutative continued fraction rearrangements. As a consequence, the solution is a positive Laurent polynomial of the seed data.