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Abstract: In previous work, we showed that the solution of certain systems of discreteintegrable equations, notably $Q$ and $T$-systems, is given in terms ofpartition functions of positively weighted paths, thereby proving the positiveLaurent phenomenon of Fomin and Zelevinsky for these cases. This method ofsolution is amenable to generalization to non-commutative weighted paths. Undercertain circumstances, these describe solutions of discrete evolution equationsin non-commutative variables: Examples are the corresponding quantum clusteralgebras BZ, the Kontsevich evolution DFK09b and the $T$-systems themselvesDFK09a. In this paper, we formulate certain non-commutative integrableevolutions by considering paths with non-commutative weights, together with anevolution of the weights that reduces to cluster algebra mutations in thecommutative limit. The general weights are expressed as Laurent monomials ofquasi-determinants of path partition functions, allowing for a non-commutativeversion of the positive Laurent phenomenon. We apply this construction to theknown systems, and obtain Laurent positivity results for their solutions interms of initial data.

Autor: Philippe Di Francesco, Rinat Kedem


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