Start with the really well-known fact that $R[x_1, \ldots, x_n]^{S_n}$, where $R$ is any commutative ring, is polynomial on elementary symmetric polynomials. Now consider the slight generalization of multiple collections of variables, namely $R[x(i)_1, \ldots, x(i)_n]^{S_n}$, where $i$ runs over some finite indexing set and $S_n$ still acts by permuting subscripts. These rings are generally not polynomial algebras, in particular when $R$ is ${\mathbb F}_p$.

Ten years ago, in the context of computing the cohomology of symmetric groups, Mark Feshbach gave generators and inductively-defined relations for these rings when $R$ is ${\mathbb F}_2$. My questions are:

(1) Does anyone know of calculations over ${\mathbb F}_p$ or other approaches over ${\mathbb F}_2$?

(2) Restricting to $R = {\mathbb F}_p$ and replacing $S_n$ by $GL_n({\mathbb F}_p)$ we get the Dickson algebras in the case of one collection of variables. Has anyone studied the analogues of Dickson algebras where there are multiple collections of variables?