# The Isomorphism Problem for cominuscule Schubert Varieties

## TL;DR

This paper characterizes when Schubert varieties in different cominuscule flag varieties are isomorphic, extending the classification from Grassmannians to all cominuscule types through a type-independent proof.

## Contribution

It provides a type-independent criterion for isomorphism of Schubert varieties based on labeled poset isomorphisms, generalizing previous results for Grassmannians.

## Key findings

- Schubert varieties are isomorphic iff their labeled posets are isomorphic
- Generalization of Grassmannian classification to all cominuscule flag varieties
- Type-independent proof of the isomorphism criterion

## Abstract

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent.