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Bergman Complexes, Coxeter Arrangements, and Graph Associahedra

Federico Ardila·Victor Reiner·Lauren Williams
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Abstract

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 Phi with corresponding oriented matroid M_Phi, then B+(M_Phi) is dual to the graph associahedron of type Phi, and B(M_Phi) equals the nested set complex of A. In addition, we prove that for any orientable matroid M, one can find |mu(M)| different reorientations of M such that the corresponding positive Bergman complexes cover B(M), where mu(M) denotes the Mobius function of the lattice of flats of M.

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Federico Ardila, Victor Reiner, Lauren Williams. 2005-09-14. Bergman Complexes, Coxeter Arrangements, and Graph Associahedra. https://arxiv.org/abs/math/0508240

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