On prehomogeneity of a rank variety

Abstract

If a linear algebraic group G acts on M (m, n), then it also acts on a rank variety M (r) (m, n) = {X ∈ M (m, n)| 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 C.