Positroid cluster structures from relabeled plabic graphs

TL;DR

This paper explores how relabeling boundary vertices of plabic graphs induces new cluster structures on positroid varieties, revealing deep connections and isomorphisms between different varieties and their cluster seeds.

Contribution

It demonstrates that relabeling plabic graphs can produce new cluster structures on positroid varieties and establishes isomorphisms between these varieties via twist maps.

Findings

Relabeling boundary vertices yields new cluster structures on positroid varieties.

The varieties $P_v$ and $P_w$ are isomorphic through a twist isomorphism.

The conjecture that all related seeds are connected by mutations and Laurent transformations is supported for Schubert varieties.

Abstract

The Grassmannian is a disjoint union of open positroid varieties $P_v$, certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring of $P_v$ is a cluster algebra, and each reduced plabic graph $G$ for $P_v$ determines a cluster. We study the effect of relabeling the boundary vertices of $G$ by a permutation $r$. Under suitable hypotheses on the permutation, we show that the relabeled graph $G^r$ determines a cluster for a different open positroid variety $P_w$. As a key step of the proof, we show that $P_v$ and $P_w$ are isomorphic by a nontrivial twist isomorphism. Our constructions yield many cluster structures on each open positroid variety $P_w$, given by plabic graphs with appropriately relabeled boundary. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and Laurent monomial transformations involving frozen variables, and establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs $G$, the "source" cluster and the "target" cluster are related by mutation and Laurent monomial rescalings.