Positively oriented matroids are realizable

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.

Realizable but not Strongly Euclidean Oriented Matroids

The extension space conjecture of oriented matroid theory claims that the space of all (non-zero, non-trivial, single-element) extensions of a re-alizable oriented matroid of rank r is homotopy equivalent to an (r ? 1)-sphere. In 1993, Sturmfels and Ziegler proved the conjecture for the class of strongly Euclidean oriented matroids, which includes those of rank at most 3 or corank at most 2. Th...

Complete enumeration of small realizable oriented matroids

Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures, i.e., the underlying oriented matroids up to isomorphism, are often more important than their metric structures. For example, the convexity, the face lattice of the convex hull and all possible triangulations of a given point configurat...

Positroids, non-crossing partitions, and positively oriented matroids

We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, an...

Fachbereich 3 Mathematik Oriented Matroids Oriented Matroids