Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Grassmann-Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of decomposable vectors in $\wedge^d V$ (i.e. which are of the form $v_1 \wedge ... \wedge v_d$), thus describing the ideal corresponding to the Plücker embedding $\text{Gr}_d(V) \to \mathbb{P}(\wedge^d V)$.

In other words, Given an $d$ by $n$ matrix these are the relations between the determinants of $d$ by $d$ minors of the matrix.

What is known about the variety of decomposable elements for the **symmetric power** instead of the exterior power? (The case where $K$ is the field of complex numbers is sufficiently interesting.) In other words, Given an $d$ by $n$ (say, comlplex) matrix what can be said about the relations between the ${{n+d-1} \choose {d}}$ permanents of $d$ by $d$ (multi) submatrices of the matrix (we allow repeated columns but not repeated rows)?

I made a quick Google search and I found one possibly-related paper entitled "permanental ideals" by Reinhard C. Laubenbacher and Irena Swanson. They point out that here there is a strong dependence on the characteristics. (I am not sure this is indeed related but this was the closeset I found.)