TY - JOUR

T1 - The Non-Commutative A 1 T-System and its Positive Laurent Property

AU - Di Francesco, Philippe

N1 - Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.

PY - 2015/4

Y1 - 2015/4

N2 - We define a non-commutative version of the A1 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 A1. We solve the system by generalizing the flat GL2 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.

AB - We define a non-commutative version of the A1 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 A1. We solve the system by generalizing the flat GL2 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.

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U2 - 10.1007/s00220-014-2223-6

DO - 10.1007/s00220-014-2223-6

M3 - Article

AN - SCOPUS:84925465067

VL - 335

SP - 935

EP - 953

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -