Decomposition theorems for unmatchable pairs in groups and field extensions

TL;DR

This paper develops structural theorems characterizing unmatchable pairs in abelian groups and field extensions, revealing deep parallels and providing new criteria and guarantees for matchability and unmatchability.

Contribution

It introduces parallel structure theorems that characterize unmatchable pairs, unifying group and field extension frameworks with new insights and criteria.

Findings

Reveals structural obstructions to matchability in groups and fields.

Provides short proofs of known results using the new framework.

Establishes existence of nontrivial unmatchable pairs.

Abstract

A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for homogeneous polynomials, has since been extended to the setting of field extensions and to that of matroids. Earlier approaches have produced numerous criteria for matchability and unmatchability, but have offered little structural insight. In this paper, we develop parallel structure theorems which characterize unmatchable pairs in both abelian groups and field extensions. Our framework reveals analogous obstructions to matchability: nearly periodic decompositions of sets in the group setting correspond to decompositions of subspaces involving translates of a subfield in the linear setting. This perspective not only recovers previously known results through short proofs, but also leads to new matching criteria and guarantees the existence of nontrivial unmatchable pairs.