Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rotationally symmetric plabic graphs and the Lagrangian Grassmannian

Olha Shevchenko

TL;DR

The paper defines the totally nonnegative Lagrangian Grassmannian LG^{R}_{ geq 0}(n,2n), consisting of n-dimensional subspaces isotropic with respect to a skew-symmetric bilinear form R, and develops a symmetry-enforced combinatorial framework using rho-symmetric, non-reduced plabic graphs. It characterizes the cell decomposition indexed by rho-symmetric bounded affine permutations f in B^{R}(n,2n), proving each cell Pi^{>0,R}_{f} is homeomorphic to R^{d_{R}(f)} and that closure relations correspond to the symmetric affine Bruhat order f ⪯_{R} g; moreover, LG^{R}_{ geq 0}(n,2n) is a closed ball with dimension equal to {n+1 choose 2}. The work also provides a Plücker-minor description of LG^{R}(n,2n) via Delta_I(X) = Delta_{R(I)}(X) and relates LG^{R}_{ geq 0}(n,2n) to symmetric Persymmetric matrices, along with a thorough development of rho-symmetric graphs, bridges, lollipops, and fixed-point insertions to parametrize cells, including a detailed analysis of when rho-symmetric graphs can be reduced. Overall, the paper extends the Plücker-graph dictionary to the symmetric Lagrangian setting and establishes a robust combinatorial/topological picture for LG^{R}_{ geq 0}(n,2n).

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.

Rotationally symmetric plabic graphs and the Lagrangian Grassmannian

TL;DR

The paper defines the totally nonnegative Lagrangian Grassmannian LG^{R}_{ geq 0}(n,2n), consisting of n-dimensional subspaces isotropic with respect to a skew-symmetric bilinear form R, and develops a symmetry-enforced combinatorial framework using rho-symmetric, non-reduced plabic graphs. It characterizes the cell decomposition indexed by rho-symmetric bounded affine permutations f in B^{R}(n,2n), proving each cell Pi^{>0,R}_{f} is homeomorphic to R^{d_{R}(f)} and that closure relations correspond to the symmetric affine Bruhat order f ⪯_{R} g; moreover, LG^{R}_{ geq 0}(n,2n) is a closed ball with dimension equal to {n+1 choose 2}. The work also provides a Plücker-minor description of LG^{R}(n,2n) via Delta_I(X) = Delta_{R(I)}(X) and relates LG^{R}_{ geq 0}(n,2n) to symmetric Persymmetric matrices, along with a thorough development of rho-symmetric graphs, bridges, lollipops, and fixed-point insertions to parametrize cells, including a detailed analysis of when rho-symmetric graphs can be reduced. Overall, the paper extends the Plücker-graph dictionary to the symmetric Lagrangian setting and establishes a robust combinatorial/topological picture for LG^{R}_{ geq 0}(n,2n).

Abstract

We introduce the totally nonnegative Lagrangian Grassmannian , a new subset of the totally nonnegative Grassmannian consisting of subspaces isotropic with respect to a certain bilinear form . 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.

Paper Structure

This paper contains 31 sections, 50 theorems, 148 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Symmetric graphs are not always reduced
  4. $\mathop{\mathrm{LG^{R}_{\geq 0}}}\nolimits (n,2n)$ through totally nonnegative symmetric matrices and Plücker coordinates
  5. Background
  6. Totally nonnegative Grassmannian
  7. Permutations and positroid cells
  8. Bruhat order on permutations
  9. Plabic graphs
  10. Adding bridges
  11. Adding lollipops
  12. Grassmannian necklace
  13. The totally nonnegative Lagrangian Grassmannian
  14. A Plücker description of $\mathop{\mathrm{LG^{R}}}\nolimits (n,2n)$
  15. The first Schubert cell of $\mathop{\mathrm{LG^{R}_{\geq 0}}}\nolimits (n,2n)$
  16. ...and 16 more sections

Key Result

Theorem 1.2

Figures (11)

  • Figure 1: The two networks represent the same point $X$. The one on the left is reduced, but not $\rho$-symmetric, while the one on the right is $\rho$-symmetric, but not reduced. Here, unmarked edges have weight $1$.
  • Figure 2: The $\rho$-symmetric graphs parametrizing the top cells for $n=2$ and $3$.
  • Figure 3: Some local moves for plabic graphs.
  • Figure 4: Strand permutation for a reduced plabic graph.
  • Figure 5: The process of adding a bridge between $i$ and $i+1$, white at $i$. We add a new edge on the first step and then adjust the colors to make sure the resulting graph is bipartite on the second step.
  • ...and 6 more figures

Theorems & Definitions (123)

  • Theorem 1.2
  • Example 1.3
  • Example 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5: postnikov, Theorem 17.8
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8: postnikov, Theorem 4.8
  • ...and 113 more