Combinatorial Inequalities for Geometric Lattices

Citation

Stonesifer, John Randolph (1973) Combinatorial Inequalities for Geometric Lattices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7z4d-0j63. https://resolver.caltech.edu/CaltechTHESIS:06272025-201557863

Abstract

A geometric lattice is a semimodular point lattice L. The i^th Whitney number of Lis the number of elements of rank i in L. The logarithmic concavity conjecture states that

W_i(L)²/W_i-1(L)W_i+1(L) ≥ 1

for any finite geometric lattice L.

In a finite nondirected graph without loops or double edges, a set of edges is closed if whenever it contains all but one edge of a cycle, it contains the whole cycle. With set containment as the order relation, the closed sets of such a graph form a geometric lattice. It is shown that any such lattice satisfies the first nontrivial case of the logarithmic concavity conjecture. In fact,

W₂(L)²/W₁(L)W₃(L) ≥ 3/2 · (W₁(L)-1)/(W₁(L)-2) ·

This is a best possible result since equality holds for graphs without cycles.

The cut-contraction of a geometric lattice L with respect to a modular cut Q of L is the geometric lattice L - T where T = {x Є L : x Є Q, Ǝq Є Q Э x q}. It is shown that any geometric lattice L can be obtained from the Boolean algebra with W₁(L) points by means of a sequence of k = W₁(L) - dim(L) cut-contractions.

Thesis Files