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Parametrizing the Grassmannian using pipe dreams

Kartik Singh

TL;DR

The paper develops a pipe-dream–style parametrization for Deodhar components of the Grassmannian Gr_{k,n} by constructing restricted-path networks from Go-diagrams. It provides a dual framework to realize Gr_{n−k,n} via dual restricted paths, and enriches Plücker coordinates by attaching summands to restricted paths and employing toggling moves on Go-diagrams. This approach yields a practical, product-structure description of points in Deodhar components and clarifies how Deodhar components sit inside closures of others. It also offers a systematic method to study boundary behavior and closure relations within the Deodhar decomposition, with potential generalizations and conjectures guiding further work.

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.

Parametrizing the Grassmannian using pipe dreams

TL;DR

The paper develops a pipe-dream–style parametrization for Deodhar components of the Grassmannian Gr_{k,n} by constructing restricted-path networks from Go-diagrams. It provides a dual framework to realize Gr_{n−k,n} via dual restricted paths, and enriches Plücker coordinates by attaching summands to restricted paths and employing toggling moves on Go-diagrams. This approach yields a practical, product-structure description of points in Deodhar components and clarifies how Deodhar components sit inside closures of others. It also offers a systematic method to study boundary behavior and closure relations within the Deodhar decomposition, with potential generalizations and conjectures guiding further work.

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 . 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.

Paper Structure

This paper contains 13 sections, 22 theorems, 71 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Partitions, Go-diagrams and L-diagrams
  4. Grassmannian and Schubert decomposition
  5. Matroid decomposition
  6. Positroid decomposition
  7. Deodhar decomposition
  8. Talaska--Williams parametrization of the Grassmannian
  9. Restricted path parametrization
  10. Dual restricted paths
  11. Plücker coordinates
  12. Closure of Deodhar components
  13. Generalizing \ref{['theorem:closurerelation']}

Key Result

Theorem 2.12

Let $\textup{v}$ be a distinguished subexpression of $\textup{w}$. Then, $\mathcal{R}_{\textup{v},\textup{w}} = G_{\textup{v},\textup{w}}B^+$.

Figures (22)

  • Figure 1: Attaching transpositions and an example for a reading word
  • Figure 2: Possible cases at a cell. The lower index pipe is coloured in red.
  • Figure 3: Example of a pipe dream which is not a Go-diagram.
  • Figure 4: Example of a pipe dream, which is a Go-diagram.
  • Figure 5: Go-diagram and its corresponding network $M_D$.
  • ...and 17 more figures

Theorems & Definitions (74)

  • Definition 2.1: Distinguished subexpressions
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8: Matroid stratum
  • Remark 2.9
  • Remark 2.10
  • ...and 64 more