Bergman Complexes, Coxeter Arrangements, and Graph Associahedra

TL;DR

This paper explores the relationships between Bergman complexes, Coxeter arrangements, and graph associahedra, revealing dualities and covering properties that deepen the understanding of tropical varieties in algebraic geometry.

Contribution

It establishes a duality between positive Bergman complexes and graph associahedra for Coxeter arrangements, and shows how reorientations of matroids can cover their Bergman complexes.

Findings

B+(M_Phi) is dual to the graph associahedron of type Phi.

B(M_Phi) equals the nested set complex of the arrangement.

Reorientations of a matroid can cover its Bergman complex.

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 Mobius function of the lattice of flats of M.