Inequalities for the h-Vectors 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 Δ(L) is the order complex of a rank (r + 1) geometric lattice L, then for all i ≤ r/2 the h-vector of Δ(L) satisfies hi-1 ≤ hi and hi ≤ hr-i. We also obtain several inequalities for the flag h-vector of Δ(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)).