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Q-Systems, Heaps, Paths and Cluster Positivity

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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. Considered as a discrete integrable dynamical system, we show that the conserved quantities are partition functions of hard particles on certain weighted graphs determined by the choice of initial data. This allows us to interpret the solutions of the system as partition functions of Viennot’s heaps on these graphs, or as partition functions of weighted paths on dual 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 the relation to domino tilings of deformed Aztec diamonds with defects.

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Authors and Affiliations

  1. Institut de Physique Théorique du Commissariat à l’Energie Atomique, Unité de Recherche Associée du CNRS, CEA Saclay/IPhT/Bat 774, F-91191, Gif sur Yvette Cedex, France

    Philippe Di Francesco

  2. Department of Mathematics, University of Illinois MC-382, Urbana, IL, 61821, U.S.A.

    Rinat Kedem

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  1. Philippe Di Francesco
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  2. Rinat Kedem
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Correspondence to Rinat Kedem.

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Communicated by L. Takhtajan

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Francesco, P.D., Kedem, R. Q-Systems, Heaps, Paths and Cluster Positivity. Commun. Math. Phys. 293, 727–802 (2010). https://doi.org/10.1007/s00220-009-0947-5

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  • DOI: https://doi.org/10.1007/s00220-009-0947-5

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