Séminaire Lotharingien de Combinatoire, 80B.44 (2018), 12 pp.
Felix Gotti
On Positroids Induced by Rational Dyck Paths
Abstract.
A rational Dyck path of type (m,d) is an increasing unit-step
lattice path from (0,0) to (m,d) in Z2 that never goes above
the diagonal line y = (d/m)x. On the other hand, a positroid of rank
d on the ground set [d+m] is a special type of matroid coming from
the totally nonnegative Grassmannian. In this paper we describe how to
naturally assign a rank d positroid on the ground set [d+m], which
we name rational Dyck positroid, to each rational Dyck path of
type (m,d). Positroids can be parameterized by several families of
combinatorial objects. Here we characterize some of these families for
the positroids we produce, namely, decorated permutations,
Le-diagrams, and move-equivalence classes of plabic
graphs. Finally, we describe the matroid polytope of a given rational
Dyck positroid.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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