Séminaire Lotharingien de Combinatoire, B54Aj (2006), 25 pp.

Federico Ardila, Victor Reiner and Lauren Williams

Bergman Complexes, Coxeter Arrangements, and Graph Associahedra

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 Möbius function of the lattice of flats of M.

Received: September 14, 2005. Accepted: December 11, 2006.

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