On the possible dimensions of subspace intersections

Abstract

A problem concerning the dimensions of the intersections of a subspace in the direct sum of a finite number of the finite-dimensional vector spaces with the pairwise sums of direct summands, provided that the subspace intersection with these direct summands is zero has been discussed. The problem is naturally divided into two ones, namely, the existence and representability of the corresponding matroid. Necessary and sufficient conditions of the existence of a matroid with the specified ranks of some subsets of the base set have been given. Using these conditions, necessary conditions of the existence of a matroid with a base set composed of a finite series of pairwise disjoint sets of the full rank and the given ranks of their pairwise unions have been presented. A simple graphical representation of the latter conditions has also been considered. These conditions are also necessary for the subspace to exist. At the end of the paper, a conjecture that these conditions are sufficient has also been stated.