On the combinatorics of Galois numbers

Ulrich Faigle asked whether this has an immediate combinatorial interpretation in the following sense: Is it always possible to partition the lattice of subspaces of GF (q) into two intervals of length n− 1 and (qn−1 − 1) intervals of length n− 2, for n ≥ 2? We show that such an interval decomposition of the lattice of subspaces of a finite-dimensional vector space exists if and only if there exists a family of linear forms with certain properties. As applications we prove that all finite-dimensional real vector spaces admit an interval decomposition while GF (2) has an interval decomposition if and only if n ≤ 4. This partially answers the question of Faigle to the negative. On the other hand we present an interval decomposition of GF (3) and further results we found using a computer.