Two statements on path systems related to quantum minors

TL;DR

This paper proves two key assertions about path matrices derived from planar directed graphs, linking minors to disjoint path systems and analyzing transformations of flow pairs, advancing combinatorial understanding of quantum minors.

Contribution

It establishes that minors of path matrices are determined by disjoint path systems and analyzes transformations of flow pairs, completing previous unproved statements.

Findings

Minors of path matrices are determined by disjoint path systems.

Transformations of flow pairs are characterized and analyzed.

Completes proofs of previously stated assertions about path matrices.

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 $G$.