Q-Systems, heaps, paths and cluster positivity

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)727-802
Number of pages76
JournalCommunications in Mathematical Physics
Volume293
Issue number3
DOIs
StatePublished - Dec 1 2009

Fingerprint

Heap
Positivity
Partition Function
partitions
Path
Domino Tilings
Positive Polynomials
Dual Graph
Cluster Algebra
Lattice Paths
Laurent Polynomials
Conserved Quantity
Weighted Graph
mutations
Continued fraction
Strombus or kite or diamond
Rearrangement
General Solution
dynamical systems
Generating Function

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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

In: Communications in Mathematical Physics, Vol. 293, No. 3, 01.12.2009, p. 727-802.

Research output: Contribution to journalArticle

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