Dense PGL-orbits in products of Grassmannians

In this paper, we find some necessary and sufficient conditions on the dimension vector $\underset{‾}{\mathbf{d}}=(d_1,\mathrm{.}\mathrm{.}\mathrm{.},d_k;n)$ 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 $(d,d,\mathrm{.}\mathrm{.}\mathrm{.},d;n)$.