We study highest weight vectors for symmetric and alternating spaces of tensors, whose dimensions are given by generalized Kronecker coefficients. We present a unified explicit construction for corresponding spanning sets of highest weight vectors and completely describe the linear relations among them. We prove that these highest weight vectors satisfy a natural yet nontrivial duality. As applications, we also give conceptual interpretations to power expansions of Cayley's first hyperdeterminant and its dual exterior Cayley form.