### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 727-802 |

Number of pages | 76 |

Journal | Communications in Mathematical Physics |

Volume | 293 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2009 |

### Fingerprint

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Q-Systems, heaps, paths and cluster positivity.** / Di Francesco, Philippe; Kedem, Rinat.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 293, no. 3, pp. 727-802. https://doi.org/10.1007/s00220-009-0947-5

}

TY - JOUR

T1 - Q-Systems, heaps, paths and cluster positivity

AU - Di Francesco, Philippe

AU - Kedem, Rinat

PY - 2009/12/1

Y1 - 2009/12/1

N2 - 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. 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 ArQ-system. We also give the relation to domino tilings of deformed Aztec diamonds with defects.

AB - 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. 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 ArQ-system. We also give the relation to domino tilings of deformed Aztec diamonds with defects.

UR - http://www.scopus.com/inward/record.url?scp=71249136740&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71249136740&partnerID=8YFLogxK

U2 - 10.1007/s00220-009-0947-5

DO - 10.1007/s00220-009-0947-5

M3 - Article

AN - SCOPUS:71249136740

VL - 293

SP - 727

EP - 802

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -