$\PGL$ orbits on products of flag varieties

TL;DR

This paper investigates conditions under which the diagonal action of PGL(n) on products of partial flag varieties admits a dense orbit, identifying specific cases where dense orbits exist or do not exist.

Contribution

It characterizes when dense orbits occur for certain products of flag varieties under PGL(n) action, extending understanding of orbit structure in algebraic geometry.

Findings

Dense orbits exist for flag varieties with specific parameters.

Certain infinite families of products lack dense orbits.

Conditions for existence of dense orbits are explicitly determined.

Abstract

In this paper, we study the existence of a dense orbit for the diagonal $\PGL(n)$ action on self-products of partial flag varieties. We determine when there exists a dense orbit for flag varieties of the form $F(k_1, \dots, k_r; n)^m$ when $k_i = i$ or when $m=3$ and $k_i = \ell + i$ for some $\ell \geq 0$. We also show that certain infinite families of products do not have a dense orbit.