The Rank Function of a Positroid and Non-Crossing Partitions

Abstract

A positroid is a special case of a realizable matroid, that arose from the study of totally nonnegative part of the Grassmannian by Postnikov [13]. Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations. The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation [10]. In this paper, we show that the rank of an arbitrary set in a positroid can be computed directly from the associated decorated permutation using non-crossing partitions.

Lattice Path Matroids and Quotients

“…Combining our description of LPM quotients with the fact that LPFMs are points in F ≥0 n we achieve our final result Theorem 41: the (realizable) quotient relation among LPMs can be expressed in terms of certain objects called CCW arrows in [38]. This confirms a conjecture of Mcalmon, Oh, Xiang in the case of LPMs.…”

Combinatorics of essential sets for positroids

“…In particular, if i is a loop of the positroid, then π(i) = i, and if i is a coloop of the positroid, then π(i) = i + n. Moreover, information on the rank of a positroid on cyclic intervals can be extracted from its associated bounded affine permutation. Indeed, by specializing [11,Theorem 3] to the case of cyclic intervals, we obtain the following result. Proposition 2.7.…”

Combinatorics of the geometry of Wilson loop diagrams I: equivalence classes via matroids and polytopes

“…In this section, we present several results about matroids, transversal matroids, Rado matroids, and positroids that are needed for our results. See, for example, [26,28,15,29], Experts on the subject may skip this exposition.…”

“…The set of Grassmann necklaces is in bijection with the set of postitroids. However, one may associate a Grassmann necklace to any matroid via its set of bases, [26,29]. Specifically, for any matroid M = ([n], B), let I(B) be the lexicographically minimal element of B in the < i linear order on [n].…”

“…Definition 3.23 (Section 2.2 [26]). The Gale ordering is a partial ordering on sets of size k with elements in the cyclically ordered set [n].…”