2008/11/30 by Philippe Di Francesco, P. Di Francesco, Rinat Kedem +1 · 66 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Cluster (spacecraft) #Cluster algebra #Combinatorics #Computer science #Conjecture #Discrete mathematics #Domino #Exponential function #Lattice (music) #Mathematical analysis #Mathematics #Partition (number theory) #Physics #Pure mathematics #Random Matrices and Applications #Statistical physics #cond-mat.stat-mech #math-ph #math.CO #math.MP #math.QA #math.RT
paper · pdf · doi:10.1007/s00220-009-0947-5
published in Communications in Mathematical Physics 293(3), 727-802 (Springer Science+Business Media) · 106 pages, 38 figures
openalex publication_date 2009/11/05 · arxiv created 2010/06/24 · arxiv updated 2015/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the cluster algebra associated to the Q-system for Ar 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 Ar Q-system. We also give an alternative formulation in terms of domino tilings of deformed Aztec diamonds with defects.