Matroid polytopes, nested sets and Bergman fans

By Eva Maria Feichtner, Bernd Sturmfels. 2004

296 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/7c9db7ffc08a315c62dc1d6c4d9305ef893fdcd8

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In Algebra and Number Theory

1. T. Panov
2. Jenia Tevelev
3. Michael Joswig
4. Federico Ardila
5. V. Buchstaber
6. Jascha Smacka
7. Marcelo Aguiar
8. Sven Herrmann

In Computational Theory and Mathematics

1. Jenia Tevelev
2. Michael Joswig
3. Federico Ardila
4. V. Buchstaber
5. David E. Speyer
6. T. Panov
7. Marcelo Aguiar
8. Jascha Smacka

In Discrete Mathematics and Combinatorics

1. T. Panov
2. V. Buchstaber
3. Michael Joswig
4. David E. Speyer
5. Federico Ardila
6. Marcelo Aguiar
7. Sven Herrmann
8. Jascha Smacka

Citations in the corpus

• Hopf Monoids and Generalized Permutahedra (2017, background): “As shown in [30, 63] and illustrated in Figure 12(b), nested sets for B are in bijection with a family of objects called B-forests, as follows.”
• Superforms, tropical cohomology, and Poincaré duality (2015, background): “constructions of different polyhedral structures for this fan coming from the matroid; see [9; 3].”
• Toric Topology (2012, background): “Our exposition is much influenced by the original works of Feichtner–Sturmfels [90] and Postnikov [189].”
• Compactifications of subvarieties of tori (2004, background): “ted on X nest and gives a regular map X nest → X vc identical on X. Example 4.1 shows that X wond is not necessarily equal to X vc. The combinatorial condition to guarantee X wond = X vc was found in [FS] (geometrically it means that any strata of X wond is of log general type). §5. Tropical Quotien”