Closures of O_n orbits in the flag variety for GL_n, II

TL;DR

This paper characterizes when the closures of orbits in the flag variety for GL_n are smooth or rationally smooth using pattern avoidance criteria, linking geometric properties to combinatorial patterns.

Contribution

It provides necessary and sufficient pattern avoidance conditions for smoothness and rational smoothness of orbit closures in the flag variety for GL_n.

Findings

Pattern avoidance characterizes smooth orbit closures.

Pattern avoidance characterizes rational smoothness of closures.

Conditions are equivalent to geometric smoothness criteria.

Abstract

We give a necessary and sufficient condition in terms of pattern avoidance for the conjugates of the bottom vertex in the Bruhat graph attached to an O_n orbit O in the flag variety for GL_n to have degree equal to the rank of this graph as a poset, showing that this condition is equivalent to the rational smoothness of the closure of O. We also give a necessary and sufficient condition in terms of pattern avoidance for the closure of O to be smooth.