The twist for positroid varieties

TL;DR

This paper clarifies the relationship between two important maps related to positroid varieties, introducing a twist automorphism that provides an inverse map and Laurent formulas, advancing understanding in algebraic geometry and cluster algebras.

Contribution

It introduces a twist automorphism for open positroid varieties, connecting Postnikov's boundary measurement map with cluster structures, and provides explicit inverse and Laurent formulas.

Findings

Established a twist automorphism for positroid varieties

Connected boundary measurement maps with cluster structures

Derived Laurent formulas for twisted Pl"ucker coordinates

Abstract

The purpose of this note is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Pl\"ucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Pl\"ucker coordinates.