Discrete integrable systems are systems of recursion relations
describing evolution in a discrete time variable, with a suitable
number of independent conservation laws.
We concentrate on the examples of A1 Q- and T-systems, both part of
cluster algebras, respectively of rank 2 and infinity. As such they
enjoy the positive Laurent property: the solutions may be expressed in
terms of the initial data as Laurent polynomials with non-negative
integer coefficients.
We then formulate non-commutative analogues of these systems defined
on a non-commutative algebra A, and prove the non-commutative positive
Laurent property for their solutions. The proof relies on the
existence of a GL2(A) flat connection on the solutions of these
systems, a manifestation of their discrete non-commutative
integrability. The solutions may be interpreted combinatorially as
partition functions of paths on networks and/or dimers on graphs, with
non-commutative weights.
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