Two statements on path systems related to quantum minors

Abstract

In ArXiv:1604.00338[math.QA] we gave a complete combinatorial characterization of homogeneous quadratic identities for minors of quantum matrices. It was obtained as a consequence of results on minors of matrices of a special sort, the so-called path matrices $Path_G$ generated by paths in special planar directed graphs $G$. In this paper we prove two assertions that were stated but left unproved in ArXiv:1604.00338[math.QA]. The first one says that any minor of $Path_G$ is determined by a system of disjoint paths, called a flow, in $G$ (generalizing a similar result of Lindstr\"om's type for the path matrices of Cauchon graphs by Casteels). The second, more sophisticated, assertion concerns certain transformations of pairs of flows in $G$.