Splicing positroid varieties

TL;DR

This paper constructs an explicit isomorphism between an open subset of a positroid variety in the Grassmannian and a product of two other positroid varieties, using cluster algebra techniques.

Contribution

It introduces a new explicit isomorphism between open positroid varieties, involving freezing cluster variables and cluster quasi-equivalence.

Findings

Explicit isomorphism between open positroid varieties and product varieties.

Application of cluster algebra techniques to positroid varieties.

Enhanced understanding of the structure of positroid varieties.

Abstract

We construct an explicit isomorphism between an open subset in the open positroid variety $\Pi_{k,n}^{\circ}$ in the Grassmannian $\mathrm{Gr}(k,n)$ and the product of two open positroid varieties $\Pi_{k,n-a+1}^{\circ}\times \Pi_{k,a+k-1}^{\circ}$. In the respective cluster structures, this isomorphism is given by freezing a certain subset of cluster variables and applying a cluster quasi-equivalence.