Hilbert functions of points on Schubert varieties in Orthogonal Grassmannians

TL;DR

This paper develops a combinatorial method using standard monomial theory to compute Hilbert functions at points on Schubert varieties in orthogonal Grassmannians, extending previous work on related varieties.

Contribution

It introduces a new combinatorial approach to determine Hilbert functions for Schubert varieties in orthogonal Grassmannians, generalizing earlier methods used for other types of Grassmannians.

Findings

Provides a combinatorial formula for Hilbert functions

Interprets multiplicity as non-intersecting lattice paths

Connects geometric and combinatorial perspectives

Abstract

A solution is given to the following problem: how to compute the multiplicity, or more generally the Hilbert function, at a point on a Schubert variety in an orthogonal Grassmannian. Standard monomial theory is applied to translate the problem from geometry to combinatorics. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for the Grassmannian and the symplectic Grassmannian. As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. Taking the Schubert variety to be of a special kind and the point to be the "identity coset," our problem specializes to a problem about Pfaffian ideals treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a "generic singularity."