The Bergman complex of a matroid and phylogenetic trees

By Federico Ardila, Caroline J. Klivans. J. Comb. Theory B, 2003

316 citations recorded by the source corpus.

We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of the proper part of its lattice of flats. In addition, we show that the Bergman fan B˜(Kn) of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn × R. This leads to a proof that the link of the origin in Tn is homeomorphic to the order complex of the proper part of the partition lattice Πn.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d4ff0aaba4d8d932971f2c711feae59f8ff4b28

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Matroid Theory for Algebraic Geometers (2014, methodology): “Sturmfels gave a combinatorial definition [92] which was elaborated by Ardila-Klivans [3].”
Stellahedral geometry of matroids (2022, background): “…b was, this linear subspace is a realization of the matroid (cid:28) M = M × 0 on (cid:28) 𝐸 called the free coextension of M, whose set of bases is deﬁned as It is a classical statement [Stu02; AK06] that the tropicalization of a linear subspace is the support of the Bergman fan of the…”
Matroid polytopes, nested sets and Bergman fans (2004, background): “We examine the local structure of this triangulation in matroid-theoretic terms, thus refining the results in [ 1 ].”
Hopf Monoids and Generalized Permutahedra (2017, background): “[7] The Bergman complex of a matroid m of rank r is homeomorphic to a wedge of (−1)μ(m) spheres of dimension r − 2, where μ(m) is the Möbius number of m.”
Brief introduction to tropical geometry (2015, unknown): “1 ([AK06]).”
Tropical Discriminants (2005, unknown): “Luckily, the relevant combinatorics is well understood, thanks to [1, 10], and in the remainder of this section we summarize what is known.”
Algebraic and Geometric Methods in Enumerative Combinatorics (2014, background): “rsection with the hyperplane P i x i = 0 and the unit sphere P i x 2 i = 1; it captures all the combinatorial structure of Trop(M). Ardila and Klivans showed B(M) and ( L Mf b0;b1g) are homeomorphic. [AK06] Then: [Bjo82] The reduced Euler characteristic of the order complex ( L Mf b0;b1g) is T(1;0).”
Linear systems on tropical curves (2009, background): “A vector v ∈ TPn−1 is called primitive if it has an integral representative in Zn and generates the semigroup (Zn ∩ (R+v + R[1]))/Z[1].”
Tropical Mathematics (2004, background): “Here L X is a polyhedral fan known as the Bergman fan of a matroid [1].”
Log-concavity of characteristic polynomials and the Bergman fan of matroids (2011, background): “The Poincaré dual arises from the description of the Bergman fan studied by Ardila-Klivans [2].”
Hodge theory for combinatorial geometries (2015, background): “he same set of vertices. Twoextremecasesgivefamiliargeometricobjects. WhenP isthe setofall nonempty proper ﬂats of M, we have ΣM = ΣM,P = ΣeM,P = the ﬁne subdivision of the tropical linear space of M [AK06]. When P is empty, we have eΣ M,∅ = the r-dimensional skeleton of the normal fan of the simple”
The diagonal of tropical matroid varieties and cycle intersections (2010, methodology): “act rational tropical curves (i.e. metric trees with bounded internal edges and nunbounded labelled leaves; see [GKM, chapter 3] for the construction of M n). Our bijection is analogous to the one in [AK] except for a global scalar factor. More important, we show that this map is actually a tropical”
Superforms, tropical cohomology, and Poincaré duality (2015, background): “constructions of different polyhedral structures for this fan coming from the matroid; see [9; 3].”