Image closures of symmetric wide-matrix varieties

Let X be an affine scheme of k×N-matrices and Y be an affine scheme of N×⋯×N-dimensional tensors. The group Sym(N) acts naturally on both X and Y and on their coordinate rings. We show that the Zariski closure of the image of a Sym(N)-equivariant morphism of schemes from X to Y is defined by finitely many Sym(N)-orbits in the coordinate ring of Y. Moreover, we prove that the closure of the image of this map is Sym(N)-Noetherian, that is, every descending chain of Sym(N)-stable closed subsets stabilizes.