Delta-matroids and toric degenerations in OG(n,2n+1)

TL;DR

This paper constructs an explicit degeneration of torus orbit closures in the orthogonal Grassmannian into Richardson varieties, providing a formula for their cohomology classes and revealing a polyhedral decomposition linked to delta-matroids.

Contribution

It introduces a novel explicit degeneration of torus orbit closures in OG(n,2n+1) into Richardson varieties, connecting algebraic geometry with delta-matroid theory.

Findings

Derived a formula for the cohomology class as a sum of Schubert products.

Established a polyhedral decomposition of the hypercube via delta-matroids.

Linked moment map images to base polytopes of delta-matroids.

Abstract

We construct an explicit, embedded degeneration of the general torus orbit closure in the maximal orthogonal Grassmannian OG(n,2n+1) into a union of Richardson varieties. In particular, we deduce a formula for the cohomology class of the torus orbit closure, as a sum of products of Schubert classes. The moment map images of the degenerate pieces are the base polytopes of their underlying delta-matroids, and give a polyhedral decomposition of the unit hypercube, which had previously been studied by Chen-Sanchez-Veliz-Ying.