Inequalities for the h- and flag h-vectors of geometric lattices
Abstract
Description
We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if D(L) is the order complex of a rank (r+1) geometric lattice L, then for all i \leq r/2 the h-vector of D(L) satisfies h(i-1) \leq h(i) and h(i) \leq h(r-i).
We also obtain several inequalities for the flag h-vector of D(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 h(i-1) \leq h(i) when i \leq (2/7)(r + 5/2).
15 pages, 2 figures. Typos fixed; most notably in Table 1. A note was added regarding a solution to problem 4.6
15 pages, 2 figures. Typos fixed; most notably in Table 1. A note was added regarding a solution to problem 4.6