Cluster Duality for Lagrangian and Orthogonal Grassmannians

TL;DR

This paper establishes a duality between cluster structures and mirror symmetry for Lagrangian Grassmannians, linking Newton-Okounkov bodies with superpotentials on mirror orthogonal Grassmannians.

Contribution

It introduces a cluster seed for the Lagrangian Grassmannian and proves the Newton-Okounkov body matches a polytope from the superpotential on the mirror orthogonal Grassmannian.

Findings

Newton-Okounkov body agrees with the superpotential polytope

Cluster seed for Lagrangian Grassmannian constructed

Mirror symmetry connection established

Abstract

In [RW19] Rietsch and Williams relate cluster structures and mirror symmetry for type A Grassmannians Gr(k, n), and use this interaction to construct Newton-Okounkov bodies and associated toric degenerations. In this article we define a cluster seed for the Lagrangian Grassmannian, and prove that the associated Newton-Okounkov body agrees up to unimodular equivalence with a polytope obtained from the superpotential defined by Pech and Rietsch on the mirror Orthogonal Grassmannian in [PR13].