Representability of the Direct Sum of $q$-Matroids

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.