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Séminaire Lotharingien de Combinatoire, B54Aj (2006), 25 pp.

# Federico Ardila, Victor Reiner and Lauren Williams

# Bergman Complexes, Coxeter Arrangements, and Graph Associahedra

**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 *\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.

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