Dense PGL-orbits in products of Grassmannians

In this paper, we find some necessary and sufficient conditions on the dimension vector d̲=(d1,…,dk;n) so that the diagonal action of PGL(n) on ∏i=1kGr(di;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 classify dimension vectors where |di−dj|<3 for all i,j generalizing a theorem of Popov [11].