## Abstract

We define a non-commutative version of the A_{1} T-system, which underlies frieze patterns of the integer plane. This system has discrete conserved quantities and has a particular reduction to the known non-commutative Q-system for A_{1}. We solve the system by generalizing the flat GL_{2} connection method used in the commuting case to a 2 × 2 flat matrix connection with non-commutative entries. This allows us to prove the non-commutative positive Laurent phenomenon for the solutions when expressed in terms of admissible initial data. These are rephrased as partition functions of paths with non-commutative weights on networks, and alternatively of dimer configurations with non-commutative weights on ladder graphs made of chains of squares and hexagons.

Original language | English (US) |
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Pages (from-to) | 935-953 |

Number of pages | 19 |

Journal | Communications in Mathematical Physics |

Volume | 335 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2015 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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