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Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas)

Roger Casals, Ian Le, Melissa Sherman-Bennett, Daping Weng

Abstract

First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from a reduced plabic graph, show it is Demazure, compare their associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation.

Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas)

Abstract

First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from a reduced plabic graph, show it is Demazure, compare their associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation.
Paper Structure (46 sections, 36 theorems, 61 equations, 46 figures)

This paper contains 46 sections, 36 theorems, 61 equations, 46 figures.

Table of Contents

  1. Introduction
  2. Scientific Context
  3. Main Results
  4. Preliminaries
  5. Preliminaries on Positroids
  6. Preliminaries on Reduced Plabic Graphs
  7. Preliminaries on weaves
  8. Weaves from Reduced Plabic Graphs
  9. T-shifts of plabic graphs
  10. Positroid weaves
  11. Positroid Weaves are Complete Demazure Weaves
  12. Braid words and orientations under T-shifts
  13. An ingredient from graph theory
  14. Proof that positroid weaves are complete Demazure
  15. Weaves and Conjugate Surfaces
  16. ...and 31 more sections

Key Result

Theorem A

Let $\mathbb{G}$ be a reduced plabic graph. Then the union $\mathfrak{w}(\mathbb{G})$ of all the $T$-shifted reduced plabic graphs of $\mathbb{G}$ is a weave such that Furthermore, a variation on the moduli $\mathfrak{M}(\beta;\mathfrak{t})$, which is a cluster $\mathcal{A}$-scheme, leads to a Poisson $\mathcal{X}$-scheme $\mathfrak{M}^{\vee}(\beta;\mathfrak{t})$ such that the pair $(\mathfrak{M}

Figures (46)

  • Figure 1: (Upper left) A reduced plabic graph $\mathbb{G}$ for $\mathrm{Gr}(3,6)$. Label the boundary vertices $1, 2, \dots, 6$ clockwise, with the upper right boundary vertex labeled 1. The cluster variable for the hexagonal face $F$ is $\Delta_{135}$. (Lower left) The weave $\mathfrak{w}(\mathbb{G})$ corresponding to this plabic graph using Theorem \ref{['thm:mainA']}. The highlighted Lusztig cycle $\gamma(F)$ corresponds to the face $F$ via Theorem \ref{['thm:mainA']}.(ii). The seed $\Sigma(\mathfrak{w}(\mathbb{G}))$ is equal to $\Sigma_T(\mathbb{G})$, the target seed of the plabic graph. (Lower right) The weave $\mu_{\gamma(F)}(\mathfrak{w}(\mathbb{G}))$ obtained from mutating the weave $\mathfrak{w}(\mathbb{G})$ at $\gamma(F)$, which corresponds to a mutation of $\Sigma_T(\mathbb{G})$not represented by a plabic graph. The new cycle $\gamma(F)'$, corresponding to the vertex where we have mutated, is highlighted. The cluster variable associated to $\gamma(F)'$ is $\Delta_{136}\Delta_{245} - \Delta_{126} \Delta_{345}$, which is not a Plücker coordinate.
  • Figure 2: Example of a positive braid word $\beta_\mathcal{P}$ and its periodic grid pattern.
  • Figure 3: Rules of the road: a zig-zag strand always makes the sharpest possible at left turn at an empty vertex and makes the sharpest possible right turn at a solid vertex.
  • Figure 4: A plabic graph associated with the positroid in Example \ref{['exmp:positroid braid']}.
  • Figure 6: Example of a reduced plabic graph (left) and its quiver (right).
  • ...and 41 more figures

Theorems & Definitions (149)

  • Remark 1.1
  • Theorem A
  • Theorem B
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 139 more