On Ideal Generators for Affine Schubert Varieties

TL;DR

This paper constructs explicit generators for the ideals defining certain affine Schubert varieties and their embeddings, providing new tools for understanding their algebraic structure and applications to nilpotent orbit closures.

Contribution

It introduces explicit generators for the ideal of affine Schubert varieties via finite-dimensional embeddings, advancing the algebraic understanding of these varieties.

Findings

Explicit basis of sections for the basic line bundle constructed

Generators for the degree-one ideal part explicitly described

Conjecture on complete ideal generators for the affine Grassmannian

Abstract

We consider a certain class of Schubert varieties of the affine Grassmannian of type A. By embedding a Schubert variety into a finite-dimensional Grassmannian, we construct an explicit basis of sections of the basic line bundle by restricting certain Pl\"ucker co-ordinates. As a consequence, we write an explicit set of generators for the degree-one part of the ideal of the finite-dimensional embedding. This in turn gives a set of generators for the degree-one part of the ideal defining the affine Grassmannian inside the infinite Grassmannian which we conjecture to be a complete set of ideal generators. We apply our results to the orbit closures of nilpotent matrices. We describe (in a characteristic-free way) a filtration for the coordinate ring of a nilpotent orbit closure and state a conjecture on the SL(n)-module structures of the constituents of this filtration.