On prehomogeneity of a rank variety
On prehomogeneity of a rank variety
If a linear algebraic group $G$ acts on $M(m,n)$, then it also acts on a rank variety $M^{(r)}(m,n)=\{ X\in M(m,n)|\ \textrm {rank} X=r\}$. In this paper, we give the necessary and sufficient condition that this variety has a Zariski-dense $G$-orbit. We consider everything over the complex number field $\mathbb {C}.$