Some Properties of Simplicial Geometries




Merrick, Cynthia

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Simplicial geometries, whose points are the collection of all k-element subsets of a given (finite) ground set, were described in 1970 by Crapo and Rota [5]. So far, only geometries for which k =2 and their duals have been well-studied. In this paper, I address many general questions about simplicial geometries on n vertices, via matroid properties such as the structure of circuits, minors and orientability. I describe the smallest largely unstudied simplicial geometry, on a ground set of six vertices, with k = 3, which I call G^6_3. A construction of the only (up to symmetry) non-contractible basis is given, as well as a complete characterization of all circuits that can be built using six or fewer vertices. I prove that the matroid of G^6_3 is ternary, and give two large deleted minors which are regular. I also explore more than one method for nding topes of the associated arrangement of hyperplanes, and describe a specific construction for a simplicial tope.



Mathematics, Combinatorial geometry, Matroid, Oriented matroid, Simplicial geometry, Simplicial matroid