Characterization of rank-equality conditions for simplicial region posets

Characterize the geometric, combinatorial, or lattice-theoretic conditions on a finite central simplicial hyperplane arrangement together with a chosen base region that guarantee that the order dimension of its poset of regions equals the rank of the arrangement.

Background

For a finite central essential hyperplane arrangement, the poset of regions is ordered by inclusion of the sets of hyperplanes separating regions from a chosen base region. When the arrangement is simplicial, this poset is a lattice, and its order dimension is at least the rank of the arrangement.

The paper disproves the previously suggested general equality between order dimension and rank by exhibiting the Coxeter arrangements of types H4 and E6, whose weak orders have dimensions at least 5 and 7, respectively, exceeding ranks 4 and 6. The first open problem asks for a precise classification of the arrangements and base regions for which the rank lower bound is attained.