Ulrich and aCM bundles from invariant theory

We use certain special prehomogeneous representations of algebraic groups in order to construct aCM vector bundles, possibly Ulrich, on certain families of hypersurfaces. Among other results, we show that a general cubic hypersurface of dimension seven admitsan indecomposable Ulrich bundle of rank nine, and that a general cubic fourfold admits an unsplit aCM bundle of rank six.