Characterization of matchable sets and subspaces via Dyson transforms

TL;DR

This paper explores the structure and existence of matchings in abelian groups and field extensions, utilizing Dyson transforms to establish characterization theorems and demonstrate their applications in additive combinatorics and linear algebra.

Contribution

It introduces a linear analogue of Dyson's e-transform and provides new characterization theorems for matchable sets and subspaces in both group and linear contexts.

Findings

Characterization theorems for matchable sets in abelian groups

Linear analogue of Dyson's e-transform for subspaces

Applications demonstrating the theorems' effectiveness

Abstract

A matching from a finite subset $A$ of an abelian group $G$ to another subset $B$ is a bijection $f : A \to B$ such that $af(a) \notin A$ for all $a \in A$. The study of matchings began in the 1990s and was motivated by a conjecture of E. K. Wakeford on canonical forms for homogeneous polynomials. The theory was later extended to the linear setting of vector subspaces over field extensions, and then to matroids. In this paper, we investigate the existence and structure of matchings in both abelian groups and field extensions. Using Dyson's $e$-transform, a tool from additive combinatorics, along with a linear analogue which is introduced in this paper, we establish characterization theorems for matchable sets and subspaces. Several applications are given to demonstrate the effectiveness of these theorems as standalone tools. Throughout, we highlight the parallels between the group-theoretic and linear perspectives.