Matroid polytopes, nested sets and Bergman fans
arXiv:math/0411260
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.
22 pages, 3 figures
References in corpus (1)
Cited by in corpus (30)
- Cambrian Hopf Algebras
- The diagonal of tropical matroid varieties and cycle intersections
- Computing Tropical Linear Spaces
- Matroids from hypersimplex splits
- Matroids are not Ehrhart positive
- Geometric realizations of the accordion complex of a dissection
- Tropical curves with a singularity in a fixed point
- On the Ehrhart Polynomial of Minimal Matroids
- A-Tint: A polymake extension for algorithmic tropical intersection theory
- Hypersimplices are Ehrhart Positive
- Valuative invariants for large classes of matroids
- Moment-angle manifolds, 2-truncated cubes and Massey operations
- Which nestohedra are removahedra?
- Algorithms for Tight Spans and Tropical Linear Spaces
- Ehrhart polynomials of rank two matroids
- Chow Rings of Heavy/Light Hassett Spaces via Tropical Geometry
- Deformation cones of graph associahedra and nestohedra
- The motivic zeta functions of a matroid
- Presentations of Transversal Valuated Matroids
- Smaller extended formulations for spanning tree polytopes in minor-closed classes and beyond
- Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra
- Compactifications of subvarieties of tori
- Nested set complexes of Dowling lattices and complexes of Dowling trees
- Gorenstein matroids
- Maximum Likelihood Estimation from a Tropical and a Bernstein--Sato Perspective
- Graphs with nonnegative resistance curvature
- An infinity operad of normalized cacti
- Moduli spaces of codimension-one subspaces in a linear variety and their tropicalization
- Reconnectads
- Simplicial shellable spheres via combinatorial blowups