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Inequalities for the h and flag hvectors of geometric lattices
 Disc. and Comp. Geom
, 2003
"... Abstract. 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 hvector of ∆(L) satisfies, hi−1 ≤ hi and hi ≤ hr−i. We also obtain several inequalities f ..."
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Cited by 6 (2 self)
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Abstract. 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 hvector of ∆(L) satisfies, hi−1 ≤ hi and hi ≤ hr−i. We also obtain several inequalities for the flag hvector of ∆(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cdanalogue of the DowlingWilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag hvector proof of hi−1 ≤ hi when i ≤ 2 5 (r + 7 2). 1.
COMPLEMENTS AND HIGHER RESONANCE VARIETIES OF HYPERPLANE ARRANGEMENTS
"... Abstract. Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of t ..."
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Abstract. Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some nonlinear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using homological algebra results and vector bundles techniques, of the fact that all resonance varieties are determinantal. 1.