Representability of the direct sum of uniform q-matroids

TL;DR

This paper investigates the representability of the direct sum of uniform q-matroids, demonstrating that such sums are always representable over large fields using algebraic and geometric methods.

Contribution

It proves that the direct sum of uniform q-matroids is always representable, addressing a gap in understanding their algebraic structure.

Findings

Direct sum of uniform q-matroids is always representable.

Representation can be constructed over sufficiently large fields.

Uses algebraic and geometric tools with cyclic flats to establish results.

Abstract

There are many similarities between the theories of matroids and $q$-matroids. However, when dealing with the direct sum of $q$-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable $q$-matroids is not necessarily representable. In this work, we focus on the direct sum of uniform $q$-matroids. Using algebraic and geometric tools, together with the notion of cyclic flats of $q$-matroids, we show that this is always representable, by providing a representation over a sufficiently large field.