Pattern avoidance and K-orbit closures

TL;DR

This paper reviews pattern avoidance criteria used to determine smoothness and rational smoothness of symmetric subvarieties in flag varieties, providing proofs and references for classical cases.

Contribution

It consolidates existing pattern avoidance criteria for smoothness in symmetric subvarieties, including proofs and references, enhancing understanding of geometric properties.

Findings

Pattern avoidance criteria characterize smoothness of symmetric subvarieties.

The paper includes proofs for classical cases.

Provides references for further results.

Abstract

We review the various pattern avoidance criteria that have been developed for smoothness and rational smoothness of symmetric subvarieties of flag varieties in the classical cases, including some proofs and giving references for other results.