Parametrizing the Grassmannian using pipe dreams

TL;DR

This paper introduces a new network-based parametrization of Deodhar components of the Grassmannian using pipe dreams associated with Go-diagrams, facilitating computations and structural insights.

Contribution

It provides an alternative pipe dream-based parametrization for Deodhar components, enhancing computational ease and structural understanding of the Grassmannian.

Findings

Eases calculation of the Grassmannian isomorphism image

Associates pipe dreams with Plücker coordinate summands

Enables determination of component closure relations

Abstract

Postnikov gave a parametrization for the totally non-negative Grassmannian using the matroid decomposition and associating a network with \reflectbox{L}-diagrams. Talaska and Williams extend this result to the entire Grassmannian by using the Deodhar decomposition instead of the matroid decomposition, and the networks this time are associated with the generalized versions of \reflectbox{L}-diagrams, which are called Go-diagrams. We provide an alternative parametrization for the Deodhar components, this time constructing a network based on the pipe dreams associated with the Go-diagrams. This parametrization has several nice properties; for one, it allows us to easily calculate the image of a point under the isomorphism $Gr_{k,n}\simeq Gr_{n-k,n}$. The second feature of this parametrization is that if we write the Pl\"ucker coordinates using the Lindst\"orm-Gessel-Viennot (LGV) lemma in our parametrization, we can associate a pipe dream to each summand, which allows us to reveal additional structure on the summands. Finally, as an application of our parametrization, we describe a case where we can conclude whether one Deodhar component lies inside the closure of another.