Pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the flag variety

TL;DR

This paper provides a pattern avoidance criterion to classify rationally smooth orbit closures of certain symmetric subgroup actions on flag varieties, establishing their smoothness and invariance under isogeny.

Contribution

It introduces a new pattern avoidance criterion for classifying smooth orbit closures in flag varieties for specific symmetric subgroups.

Findings

Orbit closures fiber over smaller flag varieties with smooth fibers

All classified orbit closures are smooth

Classification is invariant under isogeny

Abstract

We give a pattern avoidance criterion to classify the orbits of Sp(p,C) x Sp(q,C) (resp. GL(n,C)) on the flag variety of type C_{p+q} (resp. D_n) with rationally smooth closure. We show that all such orbit closures fiber (with smooth fiber) over a smaller flag variety, and hence are in fact smooth. In addition we prove that the classification is insensitive to isogeny.