Subspace partitions of [formula omitted] containing direct sums II: General case

Abstract

A subspace partition of a finite vector space Fqn of dimension n over the field Fq with q elements is a collection Π of subspaces of Fqn consisting of subspaces with mutually zero intersection that partition the nonzero vectors in Fqn. Clearly, Π satisfies the equation ∑W∈Π(qdimW−1)=qn−1, which is called the packing condition. We say that Π contains a direct sum if there exist W1,…,Wr∈Π with W1⊕⋯⊕Wr=Fqn. Partitions with direct sums are ubiquitous in practice and form an important subfamily of the lattice of all subspace partitions of Fqn, which is a combinatorial q-analogue of the lattice of all set partitions of the set with n elements. In this paper, we show that subspace partitions with direct sums (where the number h of distinct dimensions among subspaces of Π is arbitrary) exist when their type (the multiset of subspace dimensions found in Π) is in the convex hull of certain kinds of vertices in the lattice of solutions of the packing condition. This generalizes a result in a previous paper, where we had considered subspace partitions containing at most h=2 distinct subspace dimensions. We also construct an infinite family of Frobenius subspace partitions with h distinct dimensions that cannot contain a direct sum, due to the fact that no combination of dimensions adds up to n.