Q-systems and cluster positivity

We consider the cluster algebra associated to the Q-system of type A_r , a system of discrete integrable equations occurring in the context of quantum spin chains. Cluster variables may be viewed as initial data for the Q-system.

We show that the conserved quantities of the Q-system are partition functions for hard particles on particular target graphs, determined by the initial data. This allows to interpret the fundamental solutions of the Q-system as generating functions of weighted paths on suitable dual target graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions that leave their final value unchanged. Finally, the remaining solutions of the Q-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices, namely paths with nearest neighbor exclusion rules. This displays all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the Q-system of type A_r . This is joint work with R. Kedem.