Abstract
There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.
Highlights
There are two reasonable ways to put a cluster structure on a positroid variety
The Grassmannian of k-planes in Cn admits a decomposition into open positroid varieties Π○(M), analogous to the decomposition of a semisimple Lie group into double Bruhat cells [FZ99]
Postnikov only shows that the boundary measurement map exists as a rational map, which is well defined on (R>0)Edges(G) Gauge
Summary
The Grassmannian of k-planes in Cn admits a decomposition into open positroid varieties Π○(M), analogous to the decomposition of a semisimple Lie group into double Bruhat cells [FZ99]. Postnikov [Pos06] showed that an appropriate choice of reduced graph G defines a boundary measurement map (C×)Edges(G) Gauge → Π○(M). Among other properties, this map can be used to parametrize the ‘totally positive part’ of Π○(M). Greg Muller and David E Speyer form a ‘cluster’ in a conjectural cluster structure on Π○(M) These homogeneous coordinates collectively define a rational coordinate chart, the face Plucker map: Π○(M) → CFaces(G) Scaling. Postnikov only shows that the boundary measurement map exists as a rational map, which is well defined on (R>0)Edges(G) Gauge. We obtain explicit birational inverses to these maps This abstract explains the definitions necessary to formulate all of our results in detail, and explains their consequences.
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