Invariant polynomial functions on k qudits

Abstract We study the polynomial functions on tensor states in (Cn)⊗k which are invariant under SU(n)k. We describe the space of invariant polynomials in terms of symmetric group representations. For k even, the smallest degree for invariant polynomials is n and in degree n we find a natural generalization of the determinant. For n, d fixed, we describe the asymptotic behavior of the dimension of the space of degree d invariants as k → ∞. We study in detail the space of homogeneous degree 4 invariant polynomial functions on (C2)⊗k.