Mathematics

The combinatorics of non-commutative discrete integrable systems

Philippe Di Francesco (UIUC)

Abstract: 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 $A_1$ 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 GL_2(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. (based on arXiv:0909.0615 and arXiv:1402.2851)