# Governing singularities of symmetric orbit closures

## TL;DR

This paper introduces new combinatorial tools like interval pattern avoidance and Mars-Springer ideals to classify singularities of symmetric orbit closures in flag varieties, specifically for the Levi subgroup GL_p x GL_q.

## Contribution

It develops a framework to classify singularity properties of symmetric orbit closures using interval patterns of clans, advancing understanding in geometric representation theory.

## Key findings

- Singularity properties can be characterized by interval patterns of clans.
- The framework applies to the action of Levi subgroup GL_p x GL_q.
- Provides a combinatorial approach to geometric singularities.

## Abstract

We develop interval pattern avoidance and Mars-Springer ideals to study singularities of symmetric orbit closures in a flag variety. This paper focuses on the case of the Levi subgroup GL_p x GL_q acting on the classical flag variety. We prove that all reasonable singularity properties can be classified in terms of interval patterns of clans.