Results and questions on matchings in groups and vector subspaces of fields

TL;DR

This paper investigates matchings in abelian groups and vector subspaces of fields, providing classifications and conditions for acyclic matchings, and explores their properties with numerous open questions.

Contribution

It offers a complete classification of field extensions with the linear acyclic matching property and analyzes the existence of acyclic matchings in finite cyclic groups.

Findings

Classified field extensions with linear acyclic matching property

Established existence results for acyclic matchings in cyclic groups

Discussed the analogy between matchings in groups and field extensions

Abstract

A matching from a finite subset $A$ of an abelian group to another subset $B$ is a bijection $f:A\rightarrow B$ with the property that $a+f(a)$ never lies in $A$. A matching is called acyclic if it is uniquely determined by its multiplicity function. Motivated by a question of E. K. Wakeford on canonical forms for symmetric tensors, the study of matchings and acyclic matchings in abelian groups was initiated by C. K. Fan and J. Losonczy in [16, 26], and was later generalized to the context of vector subspaces in a field extension [13, 1]. We discuss the acyclic matching and weak acyclic matching properties and we provide results on the existence of acyclic matchings in finite cyclic groups. As for field extensions, we completely classify field extensions with the linear acyclic matching property. The analogy between matchings in abelian groups and in field extensions is highlighted throughout the paper and numerous open questions are presented for further inquiry.