Bergman Complexes, Coxeter Arrangements, and Graph Associahedra

Federico Ardila   ; Victor Reiner   ; Lauren Williams

Séminaire lotharingien de combinatoire, 54A (2005-2007)

Résumé

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.