Dense PGL-orbits in products of Grassmannians

TL;DR

This paper characterizes when the diagonal action of PGL(n) on products of Grassmannians has dense orbits, providing algorithms and classifications for dense dimension vectors, especially with small size or uniform dimensions.

Contribution

It establishes necessary and sufficient conditions for dense orbits in products of Grassmannians and introduces algorithms for classifying dense and sparse dimension vectors.

Findings

Derived conditions for dense PGL(n)-orbits

Algorithms for classifying dimension vectors

Complete classification for small size or uniform dimensions

Abstract

In this paper, we find some necessary and sufficient conditions on the dimension vector $\underline{\bf{d}} = (d_1,..., d_k; n)$ so that the diagonal action of $\mathbb{P}GL(n)$ on $\prod_{i=1}^k Gr(d_i;n)$ has a dense orbit. Consequently, we obtain some algorithms for finding dense and sparse dimension vectors and classify dense dimension vectors with small length or size. We also characterize the dense dimension vectors of the form $(d,d,..., d; n)$.