# Smoothness of Schubert varieties indexed by involutions in finite simply laced types

## TL;DR

This paper proves that in finite simply laced types, Schubert varieties indexed by involutions are singular unless they are the longest element of a parabolic subgroup.

## Contribution

It establishes a complete characterization of singular Schubert varieties indexed by involutions in finite simply laced types.

## Key findings

- Schubert varieties indexed by involutions are singular unless they are the longest element of a parabolic subgroup
- Provides a classification of singularities in Schubert varieties based on involution indexing
- Advances understanding of geometric properties of Schubert varieties in algebraic geometry

## Abstract

We prove that in finite, simply laced types, every Schubert variety indexed by an involution which is not the longest element of some parabolic subgroup is singular.