# The Non-Commutative $A_1$ $T$-system and its positive Laurent property

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$\times$2 flat matrix connection with non-commutative entries. This allows 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.
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https://hal-cea.archives-ouvertes.fr/cea-01002522
Contributor : Emmanuelle de Laborderie <>
Submitted on : Friday, June 6, 2014 - 11:53:05 AM
Last modification on : Wednesday, September 12, 2018 - 2:13:56 PM

### Identifiers

• HAL Id : cea-01002522, version 1
• ARXIV : 1402.2851

### Citation

P. Di Francesco. The Non-Commutative $A_1$ $T$-system and its positive Laurent property. 2014. ⟨cea-01002522⟩

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