# The Cross-section of a Spherical Double Cone

## TL;DR

This paper investigates the structure of orbit closures under the action of SL(n) on the product of two partial flag varieties, establishing that the poset forms a lattice when the action is spherical.

## Contribution

It proves that the poset of SL(n)-orbit closures is a lattice in the case of spherical actions on the product of partial flag varieties.

## Key findings

- Poset of orbit closures forms a lattice under spherical action.
- Characterization of orbit closure structure in flag varieties.
- Extension of lattice properties to specific group actions.

## Abstract

We show that the poset of $SL(n)$-orbit closures in the product of two partial flag varieties is a lattice if the action of $SL(n)$ is spherical.