Positivity of the T-system cluster algebra

TL;DR

This paper provides a path model solution for the $A_r$ $T$-system cluster algebra variables, demonstrating positivity and expressing solutions as positive Laurent polynomials through non-intersecting paths and non-commutative weights.

Contribution

It introduces a path model solution for the $A_r$ $T$-system with generic boundary conditions, extending previous work on $Q$-systems and interpreting cluster mutations as non-commutative continued fractions.

Findings

Solutions are expressed as partition functions of non-intersecting paths.

The solutions are positive Laurent polynomials of initial data.

Cluster mutations correspond to non-commutative continued fraction rearrangements.

Abstract

We give the path model solution for the cluster algebra variables of the $A_r$ $T$-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 "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 non-commutative continued fraction rearrangements. As a consequence, the solution is a positive Laurent polynomial of the seed data.