On Positroids Induced by Rational Dyck Paths

Felix Gotti

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

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.