Inequalities for the H- and Flag H-vectors of Geometric Lattices

We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if ∆(L) is the order complex of a rank (r+1) geometric lattice L, then the for all i ≤ r/2 the h-vector of ∆(L) satisfies, hi−1 ≤ hi and hi ≤ hr−i. We also obtain several inequalities for the flag h-vector of ∆(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cd-analogue of the Dowling-Wilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag h-vector proof of hi−1 ≤ hi when i ≤ 2 7 (r + 5 2 ).