Positroids and non-crossing partitions

Abstract

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. This structural result yields several combinatorial facts about positroids. We show that the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions. We enumerate connected positroids, and show how they arise naturally in free probability. Finally, we prove that the probability that a positroid on [n] is connected equals 1/e 2 asymptotically.

Positively oriented matroids are realizable

“…Let a l be one of the given vectors which is not on their plane. By [ARW,Lemma 3.3], after possibly relabeling i, j, k, we may assume that i < j < k < l. This gives the following contradiction:…”

Wilson Loop Diagrams and Positroids

“…The results and definitions set forth in this section are not new. For a more comprehensive review of the material, see [5,12,17]. In the most abstract sense, a matroid is a set of independency data on a set.…”

“…, (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.…”

The Rank Function of a Positroid and Non-Crossing Partitions

“…The set of bases of a positroid can be described nicely from the Grassmann necklace [10], and the polytope coming from the bases can be described using the cyclic intervals [9], [1]. Non-crossing partitions were used to construct positroids from its connected components in [1]. They were also used in [8] as an analogue of the bases for electroids.…”