Classification of smooth horizontal Schubert varieties

TL;DR

This paper classifies smooth horizontal Schubert subvarieties in rational homogeneous varieties, showing they are homogeneously embedded cominuscule varieties classified by Dynkin diagram subdiagrams.

Contribution

It generalizes the classification of smooth Schubert varieties to include smooth horizontal cases in rational homogeneous varieties.

Findings

Smooth horizontal Schubert subvarieties are homogeneously embedded cominuscule varieties.

Classification is achieved via subdiagrams of Dynkin diagrams.

Extends known classifications to a broader class of varieties.

Abstract

We show that the smooth horizontal Schubert subvarieties of a rational homogeneous variety $G/P$ are homogeneously embedded cominuscule $G'/P'$, and are classified by subdiagrams of a Dynkin diagram. This generalizes the classification of smooth Schubert varieties in cominuscule $G/P$.