  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 $(d,d,..., d; n)$.
