Bergman Complexes, Coxeter Arrangements, and Graph Associahedra

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.Comment: 24 pages, 4 figures, new result and new proofs adde

Combinatorial aspects of total positivity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 115-119).In this thesis I study combinatorial aspects of an emerging field known as total positivity. The classical theory of total positivity concerns matrices in which all minors are nonnegative. While this theory was pioneered by Gantmacher, Krein, and Schoenberg in the 1930s, the past decade has seen a flurry of research in this area initiated by Lusztig. Motivated by surprising positivity properties of his canonical bases for quantum groups, Lusztig extended the theory of total positivity to arbitrary reductive groups and real flag varieties. In the first part of my thesis I study the totally non-negative part of the Grassmannian and prove an enumeration theorem for a natural cell decomposition of it. This result leads to a new q-analog of the Eulerian numbers, which interpolates between the binomial coefficients, the Eulerian numbers, and the Narayana numbers. In the second part of my thesis I introduce the totally positive part of a tropical variety, and study this object in the case of the Grassmannian. I conjecture a tight relation between positive tropical varieties and the cluster algebras of Fomin and Zelevinsky, proving the conjecture in the case of the Grassmannian. The third and fourth parts of my thesis explore a notion of total positivity for oriented matroids. Namely, I introduce the positive Bergman complex of an oriented matroid, which is a matroidal analogue of a positive tropical variety. I prove that this object is homeomorphic to a ball, and relate it to the Las Vergnas face lattice of an oriented matroid. When the matroid is the matroid of a Coxeter arrangement, I relate the positive Bergman complex and the Bergman complex to the corresponding graph associahedron and the nested set complex.by Lauren Kiyomi Williams.Ph.D

Tropical Homology

Given a tropical variety X and two non-negative integers p and q we define homology group $H_{p,q}(X)$. We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then $\dim H_{p,q}(X)$ coincides with the Hodge number $h^{p,q}$ of a general member of the family.Comment: 39 PAGES, 1 figure, introduction expanded and references adde

Tropical eigenwave and intermediate Jacobians

Tropical manifolds are polyhedral complexes enhanced with certain kind of affine structure. This structure manifests itself through a particular cohomology class which we call the eigenwave of a tropical manifold. Other wave classes of similar type are responsible for deformations of the tropical structure. If a tropical manifold is approximable by a 1-parametric family of complex manifolds then the eigenwave records the monodromy of the family around the tropical limit. With the help of tropical homology and the eigenwave we define tropical intermediate Jacobians which can be viewed as tropical analogs of classical intermediate Jacobians.Comment: 38 pages, 8 figure

Positively oriented matroids are realizable

We prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.Comment: 20 pages, 3 figures, references adde