Representability of the direct sum of uniform q-matroids

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.

💡 Research Summary

The paper investigates the representability of direct sums of uniform q‑matroids, a problem that behaves quite differently from the classical matroid case. While the direct sum of ordinary matroids is always representable over some field, recent work has shown that the direct sum of two representable q‑matroids may fail to be representable over any field, or may require a very large field extension. The authors focus on the special family of uniform q‑matroids, denoted U_{k,n}(q), and prove that their direct sum is always representable provided the underlying field is extended sufficiently.

The authors begin by reviewing the necessary background on rank‑metric codes, q‑systems, and q‑matroids. They recall that an