The University of Michigan Combinatorics Seminar


Abstract 

We consider the cluster algebra associated to the Qsystem 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 Qsystem. We show that the conserved quantities of the Qsystem are partition functions for hard particles on particular target graphs, determined by the initial data. This allows to interpret the fundamental solutions of the Qsystem 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 Qsystem are interpreted as partition functions for strongly nonintersecting 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 Qsystem of type A_{r }. This is joint work with R. Kedem.
