On possible dimensions of subspace intersections

Abstract

We consider the problem concerning the dimensions of the intersections of a subspace in the direct sum of a finite series of finite-dimensional vector spaces with the sums of pairs of direct summands, provided that the subspace intersection with each of these direct summands is zero. The problem is naturally divided in two: Find conditions for the existence and for the representability of the corresponding matroid. In the paper, we give necessary and sufficient conditions of the existence of a matroid if some ranks of subsets of the base set are known. Using these conditions, we also present necessary conditions of the existence of a matroid with base set composed of a finite series of disjoint sets of full rank and the ranks of their pairwise unions are given. A simple graphical representation of the latter conditions is given as well. These conditions are also necessary for the subspace to exist. At the end of the paper, we state a conjecture that these conditions are sufficient as well. Refs 4. Figs 3.