Separation of variables for scalar-valued polynomials in the non-stable range

Any complex-valued polynomial on (Rn)k decomposes into an algebraic combination of O(n)-invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if n≥2k−1. We prove that the condition n≥2k−1 is not only sufficient, but also necessary for uniqueness of the separation. Moreover, we describe the structure of non-uniqueness of the separation in the boundary cases when n=2k−2 and n=2k−3. Formally, we study the kernel of a multiplication map ϕ carrying out separation of variables. We devise a general algorithmic procedure for describing Ker ϕ in the restricted non-stable range k≤n<2k−1. In the full non-stable range n<2k−1, we give formulas for highest weights of generators of the kernel as well as formulas for its Hilbert series. Using the developed methods, we obtain a list of highest weight vectors generating Ker ϕ.