Positively oriented matroids are realizable

Abstract

Dedicated to the memory of Michel Las Vergnas.Abstract. 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.

The twist for positroid varieties

“…A positroid is a matroid with a ‘totally non‐negative’ representation; that is, is the column matroid of a matrix whose maximal minors are non‐negative real. (See for other, equivalent, definitions.) In contrast with the difficult general problem of characterizing representable matroids, positroids can be explicitly classified by several equivalent combinatorial objects, which we now recall.…”

“…By Theorem 3.3, there exists a 3 × 6 matrix such that, for any I ∈ [6] 3 , the minor with columns in I is equal to D I . One such matrix is given below in (1).…”

“…Let us consider the theorem in terms of the running example of Figure 1a. The boundary measurement map D sends the edge weights in Figure 2 to the row-span of the matrix in (1). The right twist of this matrix is given below in (2).…”

Wilson Loop Diagrams and Positroids

“…2, each matroid defines a family of Grassmannians. There is a lot known about the submanifolds of Grassmannians corresponding to matroids [4]. In the Wilson loop diagram context, one expects each Wilson loop diagram to define a cycle, or a closed submanifold, of the positive Grassmannians.…”

“…, (2,7,8), (4,5,6), (4,5,7), (5,6,7), and all distinct sets formed from these by substituting 3 for 2, and 8 for 7}). with one propagator and four vertices.…”