Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positroid Links and Braid varieties

Roger Casals, Eugene Gorsky, Mikhail Gorsky, José Simental

TL;DR

The paper develops a unified framework linking braid varieties to open positroid varieties via four equivalent data types (permutations, bounded affine permutations, cyclic rank matrices, and Le diagrams). It introduces four associated braids—Richardson, juggling, Le, and matrix braids—and proves their equivalence up to destabilizations, while showing the corresponding Legendrian links are Legendrian isotopic, providing a geometric realization of positroid strata as augmentation varieties. It further relates braid varieties to open Richardson varieties and constructs brick manifolds as smooth projective SNC compactifications, connecting their homology to top-degree Khovanov–Rozansky invariants and establishing Lefschetz-type properties. The work also discusses cluster-structure conjectures, showing how Legendrian data underpins potential cluster algebra structures on braid varieties and their mutations, and highlighting the role of exact Lagrangian fillings in parametrizing cluster charts. Overall, the results illuminate deep links between combinatorics, algebraic geometry, and contact/symplectic topology in the study of positroids, Richardson varieties, and their degenerations.

Abstract

We study braid varieties and their relation to open positroid varieties. We discuss four different types of braids associated to open positroid strata and show that their associated Legendrian links are all Legendrian isotopic. In particular, we prove that each open positroid stratum can be presented as the augmentation variety for four different Legendrian fronts described in terms of either permutations, juggling patterns, cyclic rank matrices or Le diagrams. We also relate braid varieties to open Richardson varieties and brick manifolds, showing that the latter provide projective compactifications of braid varieties, with normal crossing divisors at infinity.

Positroid Links and Braid varieties

TL;DR

The paper develops a unified framework linking braid varieties to open positroid varieties via four equivalent data types (permutations, bounded affine permutations, cyclic rank matrices, and Le diagrams). It introduces four associated braids—Richardson, juggling, Le, and matrix braids—and proves their equivalence up to destabilizations, while showing the corresponding Legendrian links are Legendrian isotopic, providing a geometric realization of positroid strata as augmentation varieties. It further relates braid varieties to open Richardson varieties and constructs brick manifolds as smooth projective SNC compactifications, connecting their homology to top-degree Khovanov–Rozansky invariants and establishing Lefschetz-type properties. The work also discusses cluster-structure conjectures, showing how Legendrian data underpins potential cluster algebra structures on braid varieties and their mutations, and highlighting the role of exact Lagrangian fillings in parametrizing cluster charts. Overall, the results illuminate deep links between combinatorics, algebraic geometry, and contact/symplectic topology in the study of positroids, Richardson varieties, and their degenerations.

Abstract

We study braid varieties and their relation to open positroid varieties. We discuss four different types of braids associated to open positroid strata and show that their associated Legendrian links are all Legendrian isotopic. In particular, we prove that each open positroid stratum can be presented as the augmentation variety for four different Legendrian fronts described in terms of either permutations, juggling patterns, cyclic rank matrices or Le diagrams. We also relate braid varieties to open Richardson varieties and brick manifolds, showing that the latter provide projective compactifications of braid varieties, with normal crossing divisors at infinity.

Paper Structure

This paper contains 30 sections, 36 theorems, 111 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Scientific Context
  3. Main Results
  4. Positroid Braids and Equivalences
  5. Combinatorial Data
  6. $k$-Grassmannian permutations and positroid pairs
  7. Le diagrams
  8. Bounded affine permutations
  9. Richardson Braids
  10. Juggling Braid
  11. The braid $J_{k}^{(1)}(f)$
  12. The braid $J^{(2)}(f)$
  13. Generalized destabilizations and comparison of $\Lambda(u,w)$ and $\Lambda(f)$
  14. Le Braid
  15. Matrix Braids
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $u,w\in S_n$ be such that $u\leq w$ in the Bruhat order and $w$ is a $k$-Grassmannian permutation, $f$ a bounded affine permutation, $r$ a cyclic rank matrix, and $\reflectbox{$L$}$ a Le-diagram. Suppose that these four pieces of data represent the same positroid type. Then

Figures (17)

  • Figure 1: The interval braid $\sigma_i\sigma_{i-1}\cdots \sigma_{j}$.
  • Figure 2: The four types of combinatorial data indexing positroid braids.
  • Figure 3: In one-line notation, $w_{\lambda} = [2,5,7,1,3,4,6]$. Right labels shown in bold.
  • Figure 4: Constructing a Le diagram from a pair $(u, w).$
  • Figure 5: The wiring diagram for the Le diagram of Example \ref{['ex: markov 44']}. If $f$ is the corresponding bounded affine permutation, then $f(1), f(2) < n$, while $f(3), f(4), f(5), f(6) > n$. Moreover, to find $f(1)$ we follow the red strand starting opposite to $1$, and see that $f(1) = 4$. Similarly, $f(2) = 6$. Likewise, to find $f(3)$ we follow the blue strand starting opposite to $3$, and see that $f(3) = 1 + 6$. Similarly, $f(4) = 3+6$, $f(5) = 5+6$, $f(6) = 2+6$. Note that the only dotless row has the right label $5$: this corresponds to $5$ being the only $x \in \{1, \dots, 6\}$ satisfying $f(x) = x+6$.
  • ...and 12 more figures

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • ...and 89 more