Parametrizing the Grassmannian using pipe dreams

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 Plücker coordinates using the Lindstörm-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.