Portugaliae Mathematica   EMIS ELibM Electronic Journals PORTUGALIAE
Vol. 62, No. 4, pp. 437-468 (2005)

Previous Article

Next Article

Contents of this Issue

Other Issues

ELibM Journals

ELibM Home



Matroid polytopes, nested sets and Bergman fans

Eva Maria Feichtner and Bernd Sturmfels

Department of Mathematics, ETH Zürich,
8092 Zürich -- SWITZERLAND
E-mail: feichtne@math.ethz.ch
Department of Mathematics, UC Berkeley,
Berkeley CA 94720 -- USA
E-mail: bernd@math.berkeley.edu

Abstract: The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila--Klivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De Concini--Procesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences.

Classification (MSC2000): 52B20.; 05B35, 14D99, 52B40, 52C35.

Full text of the article:

Electronic version published on: 7 Mar 2008.

© 2005 Sociedade Portuguesa de Matemática
© 2005–2008 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition