Inequalities for the h- and flag h-vectors of geometric lattices

Details Inequalities for the h- and flag h-vectors of geometric lattices Inequalities for the h- and flag h-vectors of geometric lattices

2004-05-27

arxiv.org/abs/math/0405535v2

Discrete and Computational Geometry, Vol. 32, No. 4, Nov. 2004, pgs 533-548

Mathematics

Combinatorics

15 pages, 2 figures. Typos fixed; most notably in Table 1. A note was added regarding a solution to problem 4.6

Scientific paper

10.1007/s00454-004-1137-z

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).

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