 
Summary: arXiv:math.CO/0508240v214Sep2005
BERGMAN COMPLEXES, COXETER ARRANGEMENTS, AND GRAPH
ASSOCIAHEDRA
FEDERICO ARDILA, VICTOR REINER, AND LAUREN WILLIAMS
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 with corresponding
oriented matroid M, then B+(M) is dual to the graph associahedron of type , and
B(M) equals the nested set complex of A. In addition, we prove that for any orientable
matroid M, one can find µ(M) different reorientations of M such that the corresponding
positive Bergman complexes cover B(M), where µ(M) denotes the M¨obius function of the
lattice of flats of M.
1. Introduction
In this paper we study the Bergman complex and the positive Bergman complex of a
Coxeter arrangement. We relate them to the nested set complexes that arise in De Concini
and Procesi's wonderful arrangement models [11, 12], and to the graph associahedra intro
duced by Carr and Devadoss [8], by Davis, Januszkiewicz, and Scott [10], and by Postnikov
[19].
