On the Classification of Schubert Varieties in Partial Flag Varieties

TL;DR

The paper advances the isomorphism classification of Schubert varieties in partial flag varieties $G/P$ by (i) solving the case of minimal parabolics and providing a full classification of Schubert surfaces into seven types; (ii) establishing cohomology-based criteria that force Cartan equivalence when two Schubert varieties are isomorphic under certain setups; (iii) developing folding techniques via diagram automorphisms to generate new isomorphisms beyond equal-support cases, with explicit folding pairs among $A_{2n-1}$, $C_n$, $D_{n+1}$, $B_n$, and exceptional types; and (iv) providing a concrete upper bound (34) for the number of isomorphism classes of Schubert three-folds. The approach combines cohomology, Lie-theoretic representations, and root-system folding to extend isomorphism results from complete to partial flag varieties and to bound higher-dimensional cases, with practical impact on the combinatorial classification of Schubert varieties across types.

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.

Figures (6)

• Figure 1: Examples of Folding
• Figure 2: The Pair of Root Systems $A_n \times A_n$ and $A_n$
• Figure 3: The Pair of Root Systems $A_{2n-1}$ and $C_n$
• Figure 4: The Pair of Root Systems $D_{n+1}$ and $B_n$
• Figure 5: The Pair of Root Systems $D_4$ and $G_2$

Theorems & Definitions (65)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Classification of Schubert Surfaces
• Conjecture 1.5
• Theorem 1.6
• Theorem 1.7
• Example 2.1
• Example 2.2
• Example 2.3