On universal quadratic inequalities for minors of TNN matrices

TL;DR

This paper characterizes universal quadratic inequalities for minors of all totally nonnegative matrices, connecting combinatorics, matrix theory, and planar graph flows to identify valid inequalities.

Contribution

It provides a combinatorial characterization of quadratic inequalities valid for all TNN matrices, extending previous results on stable identities and planar graph flows.

Findings

Characterization of quadratic inequalities for TNN matrices

Connection between minors and planar graph flows

Extension of stable quadratic identities

Abstract

For positive integers $n,n'$, we give a combinatorial characterization for the set of quadratic inequalities on minors that are valid for all $n\times n'$ totally nonnegative matrices. This is obtained as a consequence from our earlier results on stable quadratic identities on minors of matrices generated by flows in planar graphs via Lindstr\"om's construction.