Positivity of the T-system cluster algebra

Abstract

We give the path model solution for the cluster algebra variables of the Ar T-system 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 “timedependent” 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 utations are interpreted as non-commutative continued fraction rearrangements. As a consequence, the solution is a positive Laurent polynomial of the seed data.