Matchings in matroids over abelian groups, II

Abstract

The concept of matchings originated in group theory to address a linear algebra problem related to canonical forms for symmetric tensors. In an abelian group $(G,+)$, a matching is a bijection $f: A \to B$ between two finite subsets $A$ and $B$ of $G$ such that $a + f(a) \notin A$ for all $a \in A$. A group $G$ has the matching property if, for every two finite subsets $A, B \subset G$ of the same size with $0 \notin B$, there exists a matching from $A$ to $B$. In prior work [5], matroid analogues of results concerning matchings in groups were introduced and established. This paper serves as a sequel, extending that line of inquiry by investigating sparse paving, panhandle, and Schubert matroids through the lens of matchability. While some proofs draw upon earlier findings on the matchability of sparse paving matroids, the paper is designed to be self-contained and accessible without reference to the preceding sequel. Our approach combines tools from both matroid theory and additive number theory.