Bridge graphs and Deodhar parametrizations for positroid varieties

TL;DR

This paper demonstrates that two different combinatorial parametrizations of positroid varieties, one via bridge graphs and the other via Deodhar components, are essentially equivalent, unifying their understanding.

Contribution

It proves the correspondence between bridge graph parametrizations and Deodhar parametrizations for positroid varieties, establishing their equivalence.

Findings

Bridge graph parametrizations correspond to Deodhar parametrizations.

Each Deodhar parametrization can be represented by a bridge graph.

The two parametrization methods are essentially the same.

Abstract

A parametrization of a positroid variety $\Pi$ of dimension $d$ is a regular map $(\mathbb{C}^{\times})^{d} \rightarrow \Pi$ which is birational onto a dense subset of $\Pi$. There are several remarkable combinatorial constructions which yield parametrizations of positroid varieties. We investigate the relationship between two families of such parametrizations, and prove they are essentially the same. Our first family is defined in terms of Postnikov's boundary measurement map, and the domain of each parametrization is the space of edge weights of a planar network. We focus on a special class of planar networks called bridge graphs, which have applications to particle physics. Our second family arises from Marsh and Rietsch's parametrizations of Deodhar components of the flag variety, which are indexed by certain subexpressions of reduced words. Projecting to the Grassmannian gives a family of parametrizations for each positroid variety. We show that each Deodhar parametrization for a positroid variety corresponds to a bridge graph, while each parametrization from a bridge graph agrees with some projected Deodhar parametrization.