Representability of the Direct Sum of $q$-Matroids

TL;DR

This paper investigates the properties of the direct sum operation in $q$-matroids, revealing that the sum of representable $q$-matroids may not be representable, and highlights open questions in the field.

Contribution

It demonstrates that the direct sum of representable $q$-matroids can be non-representable, contrasting with classical matroid theory, and raises open problems about characterizing representability.

Findings

Direct sum of representable $q$-matroids may not be representable

Highlights differences between $q$-matroid and classical matroid theory

Open question on characterizing representability of direct sums

Abstract

While there are many parallels between matroid theory and $q$-matroid theory, most notably on the level of cryptomorphisms, there are substantial differences when it comes to the direct sum. The direct sum of $q$-matroids has been introduced in the literature only recently. In this short note we show that the direct sum of representable $q$-matroids may not be representable. It remains an open question whether representability of the direct sum can be characterized by the given $q$-matroids.