相关论文: The geometry of ranked symplectic matroids

We are interested in expanding our understanding of symplectic matroids by exploring the properties of a class of symplectic matroids with a "lattice of flats". Taking a well-behaved family of subdivisions of the cross polytope we obtain a…

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…

Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the…

We describe a natural geometric relationship between matroids and underlying flag matroids by relating the geometry of the greedy algorithm to monotone path polytopes. This perspective allows us to generalize the construction of underlying…

The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total…

The configuration of a matroid $M$ is the abstract lattice of cyclic flats (flats that are unions of circuits) where we record the size and rank of each cyclic flat, but not the set. One can compute the Tutte polynomial of $M$, and stronger…

A toric variety is constructed from a lattice polytope. It is common in algebraic combinatorics to carry this way a notion of an algebraic property from the variety to the polytope. From the combinatorial point of view, one of the most…

We show how the Tutte polynomial of a matroid $M$ can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats. The results imply that the Tutte polynomial of $M$ is already determined by the…

Polymatroids can be considered as "fractional matroid" where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a…

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…

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…

Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant. In algebraic geometry, there is a more general…

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every…

Inspired by Bruggesser-Mani's line shellings of polytopes, we introduce line shellings for the lattice of flats of a matroid: given a normal complex for a Bergman fan of a matroid induced by a building set, we show that the lexicographic…

In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of…

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…

For a lattice polytope $P$, the rank of $P$ is defined by $F-(\dim P+1)$, where $F$ is the number of facets of $P$. In this paper, we study matroid polytopes with small rank. More precisely, we characterize matroid independence polytopes…

We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid…

One generalization of ordinary matroids is symplectic matroids. While symplectic matroids were initially defined by their collections of bases, there has been no cryptomorphic definition of symplectic matroids in terms of circuits. We give…

A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic form $\omega$ on $M$, and an effective Hamiltonian action of a torus $T$ on $M$, where the dimension of $T$ is half the dimension of $M$. We prove that there is a…