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arXiv:math.CO/0406116v17Jun2004 THE POSITIVE BERGMAN COMPLEX OF AN ORIENTED MATROID
 

Summary: arXiv:math.CO/0406116v17Jun2004
THE POSITIVE BERGMAN COMPLEX OF AN ORIENTED MATROID
FEDERICO ARDILA, CAROLINE KLIVANS, AND LAUREN WILLIAMS
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 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.
1. Introduction
In [2], Bergman defined the logarithmic limit-set of an algebraic variety in order to study

  

Source: Ardila, Federico - Department of Mathematics, San Francisco State University

 

Collections: Mathematics