Séminaire Lotharingien de Combinatoire, B54Aj (2006), 25 pp.
Federico Ardila, Victor Reiner and Lauren Williams
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 \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.
The following versions are available: