Q-systems, Heaps, Paths and Cluster Positivity

TL;DR

This paper connects solutions of the $Q$-system for $A_r$ to combinatorial models like heaps and paths, proving cluster positivity and providing new interpretations of solutions as partition functions.

Contribution

It introduces a combinatorial framework linking $Q$-system solutions to heaps, paths, and tilings, establishing cluster positivity for the $A_r$ case.

Findings

Cluster variables are positive Laurent polynomials of initial data.

Solutions are interpreted as partition functions for non-intersecting paths.

Cluster mutations correspond to local rearrangements of continued fractions.

Abstract

We consider the cluster algebra associated to the $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-system are partition functions for hard particles on particular target graphs with weights, which are determined by the choice of initial data. This allows us to interpret the simplest solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently 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 which leave their final value unchanged. Finally, the general solutions of the $Q$-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the $A_r$ $Q$-system. We also give an alternative formulation in terms of domino tilings of deformed Aztec diamonds with defects.