2,083 research outputs found
Topological representations of matroid maps
The Topological Representation Theorem for (oriented) matroids states that
every (oriented) matroid can be realized as the intersection lattice of an
arrangement of codimension one homotopy spheres on a homotopy sphere. In this
paper, we use a construction of Engstr\"om to show that structure-preserving
maps between matroids induce topological mappings between their
representations; a result previously known only in the oriented case.
Specifically, we show that weak maps induce continuous maps and that the
process is a functor from the category of matroids with weak maps to the
homotopy category of topological spaces. We also give a new and conceptual
proof of a result regarding the Whitney numbers of the first kind of a matroid.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic
Combinatorics, 201
Inequalities for the h- 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 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).Comment: 15 pages, 2 figures. Typos fixed; most notably in Table 1. A note was
added regarding a solution to problem 4.
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