Bergman Complexes, Coxeter Arrangements, and Graph Associahedra

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 .