Combinatorics of the Deodhar decomposition of the Grassmannian

TL;DR

This paper explores the combinatorial structure of the Deodhar decomposition of the Grassmannian, introducing corrective flips for diagram transformations, and characterizing the inclusion relations among Deodhar components.

Contribution

It extends Lam and Williams' Le-moves to non-reduced diagrams, provides criteria for closure containment, and offers an inductive characterization of Go-diagrams.

Findings

Corrective flips transform arbitrary fillings into Go-diagrams.

Closure of Deodhar components relates to diagram modifications.

No forbidden subdiagram characterization exists for Go-diagrams.

Abstract

The Deodhar decomposition of the Grassmannian is a refinement of the Schubert, Richardson, and positroid stratifications of the Grassmannian. Go-diagrams are certain fillings of Ferrers diagrams with black stones, white stones, and pluses which index Deodhar components in the Grassmannian. We provide a series of corrective flips on diagrams which may be used to transform arbitrary fillings of Ferrers shapes with black stones, white stones, and pluses into a Go-diagram. This provides an extension of Lam and Williams' Le-moves for transforming reduced diagrams into Le-diagrams to the context of non-reduced diagrams. Next, we address the question of describing when the closure of one Deodhar component is contained in the closure of another. We show that if one Go-diagram $D$ is obtained from another $D'$ by replacing certain stones with pluses, then applying corrective flips, that there is a containment of closures of the associated Deodhar components, $\overline{\mathcal{D}'} \subset \overline{\mathcal{D}}$. Finally, we address the question of verifying whether an arbitrary filling of a Ferrers shape with black stones, white stones, and pluses is a Go-diagram. We show that no reasonable description of the class of Go-diagrams in terms of forbidden subdiagrams can exist by providing an injection from the set of valid Go-diagrams to the set of minimal forbidden subdiagrams for the class of Go-diagrams. In lieu of such a description, we offer an inductive characterization of the class of Go-diagrams.