Geodesic
Bergman Complexes, Coxeter Arrangements, and Graph Associahedra
Séminaire lotharingien de combinatoire, 54A (2005-2007)

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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 Möbius function of the lattice of flats of M. 

@article{SLC_2005-2007_54A_a9,
     author = {Federico Ardila and Victor Reiner and Lauren Williams},
     title = {Bergman {Complexes,} {Coxeter} {Arrangements,} and {Graph} {Associahedra}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {54A},
     year = {2005-2007},
     url = {http://geodesic.mathdoc.fr/item/SLC_2005-2007_54A_a9/}
}
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%F SLC_2005-2007_54A_a9
Federico Ardila; Victor Reiner; Lauren Williams. Bergman Complexes, Coxeter Arrangements, and Graph Associahedra. Séminaire lotharingien de combinatoire, 54A (2005-2007). http://geodesic.mathdoc.fr/item/SLC_2005-2007_54A_a9/