Separation of Variables for Scalar-valued Polynomials in the Non-stable Range

Any complex-valued polynomial on $(\mathbb{R}^n)^k$ decomposes into an algebraic combination of $O(n)$-invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if $n \geq 2k-1$. We prove that the condition $n\geq 2k-1$ 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 $\phi$ carrying out separation of variables. We devise a general algorithmic procedure for describing Ker $\phi$ in the restricted non-stable range $k \leq n < 2k-1$. 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 $\phi$.