Matroid adjoints and the minimum rank of zero-nonzero matrix patterns

TL;DR

This paper explores the relationship between matroid adjoints and the minimum rank of zero-nonzero matrix patterns, revealing how matroid properties influence matrix rank bounds and their dependence on field representability.

Contribution

It establishes a connection between matroid adjoints and the gap between triangle number and minimum rank, and characterizes when this gap can occur based on matroid properties.

Findings

Matroid adjoints correspond to the transpose of the fundamental pattern.

The minimum rank of the fundamental pattern is unique and depends on matroid representability.

A pattern's matroid minimum rank is smaller than matrix minimum rank over all fields in specific cases.

Abstract

The problem of finding the minimum rank of a matrix with a given zero-nonzero pattern has been generalized to a class of matroids associated to the pattern. The fundamental lower bound known as the triangle number still holds in this generalized setting. But the matroid minimum rank of a pattern need not match that of its transpose. We associate to each pattern $X$ a lattice $L(X)$. We define the fundamental pattern of a matroid $M$ to be the complement of its hyperplane-point incidence pattern and note that when $X$ is the fundamental pattern of $M$, the lattice of flats of $M$ is $L(X)$. We then prove that, for every pattern $X$, the dual lattice of $L(X)$ is isomorphic to $L(X^T)$. We show that a matroid $M'$ of the same rank as $M$ is an adjoint of $M$ if and only if $M'$ is associated with the transpose of the fundamental pattern of $M$. Our main result ties together the notion of a matroid adjoint with the phenomenon of a gap between the triangle number $k$ and the matroid minimum rank of a pattern. Namely, we show that, if any matroid of rank $k$ associated with a pattern has an adjoint, then there is no such gap for the pattern's transpose. We show that the matroid of minimum rank associated with the fundamental pattern is unique. Using this, we prove that the matrix minimum rank of the fundamental pattern of a matroid over different fields depends on the representability of the matroid over those fields. This allows us to recover and improve upon a construction of Berman et al. We also give a smaller example than any previously known of a pattern with a matroid minimum rank smaller than its matrix minimum rank over every field. Finally, we establish that, for the fundamental pattern, a converse holds to our main result. In particular, a matroid with fundamental pattern $X$ has an adjoint if and only if the matroid minimum rank of $X^T$ is equal to its triangle number.