On Possible Dimensions of Subspace Intersections for Five Direct Summands

The paper considers the problem on the dimensions of intersections of a subspace in the direct sum of a finite series of finite-dimensional vector spaces with sums of pairs of direct summands, provided that the subspace intersection with each of these direct summands is trivial. The problem naturally splits into finding conditions for the existence and representability of the corresponding matroid. The following theorem is proved: If the ranks of all the unions of a series of blocks satisfying the condition on the ranks of subsets in the matroid are given and the blocks have full rank, then this partial rank function may be extended to a full rank function for all the subsets of the base set (the union of all the blocks). Necessary and sufficient conditions on the dimensions of the direct summands and intersections mentioned above for the corresponding matroid to exist are obtained in the case of five direct summands. Bibliography: 5 titles.