The Twist for Positroids

Grassmann necklaces describe combinatorially the stratification of the Grassmannian into positroid varieties. Plabic Graphs are finer combinatorial objects than Grassmann necklaces that not only index positroids but also parametrize points in positroids, through the boundary measurement map. Given a reduced plabic graph, we can assign target face labels to each face, and the boundary face labels give us the corresponding Grassmann necklace, which tells us the non-vanishing pluckers in the positroid. It is known that the face label pluckers give a cluster in the cluster structure on the Grassmannian. A natural question is, what can the internal face labels tell us geometrically? It is not generally true that these are the non-vanishing pluckers in the image of the boundary measurement map for positroids over the complex numbers. In this talk, we introduce the twist map, constructed by Muller-Speyer, which are automorphisms of the positroid variety, taking the image of the boundary measurement map to a torus defined by the non-vanishing of face label pluckers.