Rotationally symmetric plabic graphs and the Lagrangian Grassmannian

TL;DR

This paper introduces the totally nonnegative Lagrangian Grassmannian, describes its cell structure, and shows each cell can be represented by rotationally symmetric plabic graphs, developing new techniques for non-reduced graphs.

Contribution

It defines a new subset of the nonnegative Grassmannian, characterizes its cell structure, and links cells to rotationally symmetric plabic graphs with novel methods for non-reduced cases.

Findings

Defined the totally nonnegative Lagrangian Grassmannian.

Established a cell decomposition with plabic graph representations.

Developed techniques for non-reduced plabic graphs.

Abstract

We introduce the totally nonnegative Lagrangian Grassmannian $\rm{LG}_{\geq 0}^R (n,2n)$, a new subset of the totally nonnegative Grassmannian consisting of subspaces isotropic with respect to a certain bilinear form $R$. We describe its cell structure and show that each cell admits a representation by a rotationally symmetric (not necessarily reduced) plabic graph. Along the way, we develop new techniques for working with non-reduced plabic graphs.