Positroid cluster structures from relabeled plabic graphs

Fraser, Chris¹; Sherman-Bennett, Melissa²

¹Michigan State University East Lansing MI 48824, USA
²University of Michigan Ann Arbor MI 48109, USA

Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 469-513

Abstract

The Grassmannian is a disjoint union of open positroid varieties ${\Pi }_{\mu }^{\circ }$, certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring $ℂ\left[{\Pi }_{\mu }^{\circ }\right]$ is a cluster algebra, and each reduced plabic graph $G$ for ${\Pi }_{\mu }^{\circ }$ determines a cluster. We study the effect of relabeling the boundary vertices of $G$ by a permutation $\rho$. Under suitable hypotheses on the permutation, we show that the relabeled graph $G^{\rho }$ determines a cluster for a different open positroid variety ${\Pi }_{\pi }^{\circ }$. As a key step in the proof, we show that ${\Pi }_{\pi }^{\circ }$ and ${\Pi }_{\mu }^{\circ }$ are isomorphic by a nontrivial twist isomorphism. Our constructions yield a family of cluster structures on each open positroid variety, given by plabic graphs with appropriately permuted boundary labels. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and rescalings by Laurent monomials in frozen variables. We 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.

Article information

Received: 2020-08-03
Revised: 2021-09-20
Accepted: 2021-12-31
Published online: 2022-07-07

DOI: 10.5802/alco.220

Classification: 14N35, 14M17, 57T15
Keywords: Cluster algebras, Grassmannian, positroid variety, twist map, quasi-homomorphism.

License: CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights