Q-systems, Heaps, Paths and Cluster Positivity

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.

106 pages, 38 figures

References in corpus (5)

• Total positivity, Grassmannians, and networks
• Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property
• Q-systems as cluster algebras
• Proof of the combinatorial Kirillov-Reshetikhin conjecture
• Q-system Cluster Algebras, Paths and Total Positivity

Cited by in corpus (29)

• Quiver varieties and cluster algebras
• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• Cluster algebras and continued fractions
• Noncommutative integrability, paths and quasi-determinants
• 4d N=2 Gauge Theories and Quivers: the Non-Simply Laced Case
• A compendium on the cluster algebra and quiver package in sage
• Dimers and cluster integrable systems
• Hankel determinant solutions to several discrete integrable systems and the Laurent Property
• Q-system Cluster Algebras, Paths and Total Positivity
• Discrete integrable systems, positivity, and continued fraction rearrangements
• The Graham--Knuth--Patashnik recurrence: Symmetries and continued fractions
• Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective
• The solution of the $A_r$ T-system for arbitrary boundary
• Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
• Multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions
• $Y$ systems, $Q$ systems, and 4D $\mathcal{N}=2$ supersymmetric QFT
• Quantum Q systems: From cluster algebras to quantum current algebras
• Positivity and periodicity of $Q$-systems in the WZW fusion ring
• Toda Systems, Cluster Characters, and Spectral Networks
• The solution of the quantum $A_1$ T-system for arbitrary boundary
• Solutions to the ultradiscrete Toda molecule equation expressed as minimum weight flows of planar graphs
• Linear recurrence relations in $Q$-systems and difference $L$-operators
• Linear relations for Laurent polynomials and lattice equations
• Q-Systems, Factorization Dynamics, and the Twist Automorphism
• Inverse moment problem for non-Abelian Coxeter double Bruhat cells
• Toric heaps, cyclic reducibility, and conjugacy in Coxeter groups
• $\tau$-Functions, Birkhoff Factorizations and Difference Equations
• Wall's continued-fraction characterization of Hausdorff moment sequences: A conceptual proof