The Non-Commutative $A_1$ $T$-system and its positive Laurent property - Archive ouverte HAL Access content directly
Journal Articles Communications in Mathematical Physics Year : 2015

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

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Philippe Di Francesco
• Function : Correspondent author
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#### 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 x 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.

### Dates and versions

cea-01002522 , version 1 (01-12-2022)

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• HAL Id : cea-01002522 , version 1
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### Cite

Philippe Di Francesco. The Non-Commutative $A_1$ $T$-system and its positive Laurent property. Communications in Mathematical Physics, 2015, 335, pp.935-953. ⟨10.1007/s00220-014-2223-6⟩. ⟨cea-01002522⟩

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