Matroid adjoints and the minimum rank of zero-nonzero matrix patterns
Louis Deaett, Kevin Grace
TL;DR
The paper develops a lattice-theoretic framework linking matroid adjoints to the minimum rank of zero-nonzero matrix patterns. It shows that the dual of the lattice $L(\mathcal{X})$ is isomorphic to $L(\mathcal{X}^T)$ and that adjoint existence for rank-$k$ matroids associated with a pattern enforces no gap between the triangle number and the matroid minimum rank for the pattern’s transpose; conversely, a matroid has an adjoint iff $\mathrm{mr}_{\mathcal{R}}(\mathcal{X}^T)=\mathrm{tri}(\mathcal{X}^T)$. The authors establish the uniqueness of the minimum-rank matroid for the fundamental pattern, demonstrate field-dependent behavior of matrix minimum rank, and provide a converse characterization tying adjoints to the transpose phenomenon. These results unify adjoint theory with pattern transpose behavior and offer constructive insights into field representability and patterns with small matroid minimum ranks relative to their matrix counterparts.
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.