Bergman Complexes
نویسنده
چکیده
Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Φ with corresponding oriented matroid MΦ, then B (MΦ) is dual to the graph associahedron of type Φ, and B(MΦ) equals the nested set complex of A. In addition, we prove that for any orientable matroid M , one can find |μ(M)| different reorientations of M such that the corresponding positive Bergman complexes cover B(M), where μ(M) denotes the Möbius function of the lattice of flats of M .
منابع مشابه
New Light on Bergman Complexes by Decomposing Matroid Types
Bergman complexes are polyhedral complexes associated to matroids. Faces of these complexes are certain matroids, called matroid types, too. In order to understand the structure of these faces we decompose matroid types into direct summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be subdivided into the order complex of the proper part of its lattice of flats. Beyond tha...
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