On the Classification of Schubert Varieties in Partial Flag Varieties

TL;DR

This paper extends the classification of Schubert varieties from complete flag varieties to certain partial flag varieties, providing new classifications, isomorphism pairs, and bounds on the number of isomorphism classes.

Contribution

It generalizes the classification of Schubert varieties to partial flag varieties and introduces methods to identify isomorphisms and bounds.

Findings

Classified all Schubert varieties in G/P for minimal parabolic P

Identified pairs of isomorphic Schubert varieties via root system folding

Established an upper bound on the number of isomorphism classes of Schubert three-folds

Abstract

We generalize the classification of isomorphism classes of Schubert varieties in complete flag varieties G/B to a class of partial flag varieties G/P. In particular, we classify all Schubert varieties in G/P where P is a minimal parabolic subgroup and all Schubert surfaces. We also obtain several pairs of isomorphisms of Schubert varieties from folding the root system. This allows us to find an upper bound of the cardinality of isomorphism classes of Schubert three-folds.