The positive Bergman complex of an oriented matroid

We study the positive Bergman complex B+ (M) of an oriented matroid M, which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid M. The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid MI , the positive tropical variety associated to I is equal to the fan over B+ (MI ). Our main result is that a certain "fine" subdivision of B+ (M) is a geometric realization of the order complex of the proper part of the Las Vergnas face lattice of M. It follows that B+ (M) is homeomorphic to a sphere. For the oriented matroid of the complete graph Kn, we show that the face poset of the "coarse" subdivision of B+ (Kn) is dual to the face poset of the associahedron An-2, and we give a formula for the number of fine cells within a coarse cell.

• Type: Journal
• Year: 2006
• Where: EJC
• Authors: Federico Ardila, Caroline J. Klivans, Lauren Williams
• Keywords: Bergman Complex, EJC 2006, EJC 2007, Oriented Matroid, Positive Bergman Complex