Combinatorial Theory

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.

Mathematics Subject Classifications: 05B35, 94B05, 51E20

Keywords: \(q\)-matroids, representability, evasive subspaces, rank-metric codes, linear sets