Combinatorics of the Deodhar decomposition of the Grassmannian

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.