For any partition $\lambda$ we denote by $S_\lambda$ the corresponding Schur functor. Now consider $\textrm{GL}(\mathbb{C}^n)$ with its natural action on $\mathbb{C}^n$. Using character theory, one can show that for all $d$ one has $$S_d(S_2(\mathbb{C}^n))=\oplus_{\lambda} S_\lambda\mathbb{C}^n$$ where the sum is over all partitions $\lambda$ of $2d$ with at most $n$ rows. Let us think of the left hand side as the set of degree $d$ homogeneous polynomials in the entries of a symmetric $n\times n$ matrix. Is it possible to give for each $\lambda$ an explicit set of polynomials that span $S_\lambda$ on the right hand side? Has this been already worked out somewhere?
I thought about it and I only have some rather preliminary insights. If $\lambda=(2,\ldots,2)$ ($k$ colums), then $S_\lambda\mathbb{C}^n$ should be spanned by all $k\times k$ minors. I suspect that in the general case we get some sums of products of subdeterminants but I was unable to figure it out.