Reference on the $GL_n$ representation theory of $A = Sym ( \oplus_{i = 1}^m V^*)^{\Sigma_m}$.

Question

Let $V$ be the defining representation of $GL_2(\mathbb{R})$. Let $\Sigma_m$ be the permutation group, acting on $ \oplus_{i = 1}^m V^*$ by permuting coordinates, and let $GL_2(\mathbb{R})$ act by the diagonal. These actions commute.

Let $A = Sym( \oplus_{i = 1}^m V^* ) ^{\Sigma_m}$ be the ring of $\Sigma_m$ invariants of the symmetric algebra.

I am looking for a reference on the $GL_2( \mathbb{R})$ representation theory of $A$.

I am especially interested in:

1. A generating set for $A$ as a $\mathbb{R}[ GL_2(\mathbb{R})]$ algebra, or as an $\mathbb{R}$-algebra.
2. A (combinatorial?) description of a collection of basis for each irreducible representation in $A$.
3. Other notions of compact descriptions of $A$.

Thank you! Some keywords would really help.