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

TL;DR

This paper develops a new theory connecting weaves, plabic graphs, and cluster structures in positroid varieties, proving the Muller-Speyer twist map is a Donaldson-Thomas transformation and a quasi-cluster automorphism.

Contribution

It introduces a novel construction of weaves from reduced plabic graphs, establishes their Demazure property, and proves the Muller-Speyer twist map is a Donaldson-Thomas transformation.

Findings

Weaves constructed from plabic graphs are Demazure.

The Muller-Speyer twist map is shown to be the Donaldson-Thomas transformation.

Target and source labeled seeds are related by a quasi-cluster transformation.

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.