The positive Bergman complex of an oriented matroid

  • Authors:

  • Federico Ardila;Caroline Klivans;Lauren Williams

  • Affiliations:

  • Mathematical Sciences Research Institute, Berkeley, CA;Department of Mathematics and Computer Science, University of Chicago, Chicago, IL and Computing and Information Science, Cornell University, Ithaca, NY;Department of Mathematics, MIT, Cambridge, MA

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2006

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Abstract

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 with 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.