Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...,x_m)$ and $e_k(y_1,...,y_n)$, or the power sum polynomials $p_k(x_1,\ldots,x_m)$ and $p_k(y_1,\ldots,y_n)$. In this way we are handling the variables $x_i$ and $y_j$ independently. My questions is when can $f$ be expanded in terms of polynomials of $x_i$ and $y_j$ together? For instance, can $f$ be generated by functions of the $mn$ variables $s_{ij}=x_i+y_j$?

Not quite sure what you want, but the main result on symmetric functions in the variables $xy=\{x_iy_j\,\colon\,i,j\geq 1\}$ is
$$ \langle f(xy),g(x)h(y)\rangle = \langle f,g*h\rangle,\ \ (*) $$
where the scalar product on the left is defined by
$$ \langle f(x)g(y),a(x)b(y)\rangle = \langle f,a\rangle
\langle g,b\rangle, $$
where the scalar product on the right is the standard one on symmetric functions. Similarly the scalar product on the right-hand side of (*) is the standard one, and $g*h$ denotes the internal product of symmetric
functions. See *Enumerative Combinatorics*, vol. 2, Exercise 7.78.
An equivalent formulation is
$$ s_\lambda(xy) = \sum_{\mu,\nu}g_{\lambda\mu\nu} s_\mu(x)
s_\nu(y), $$
where $s_\rho$ denotes a Schur function and
$$ g_{\lambda\mu\nu} = \langle \chi^\lambda,\chi^\mu\chi^\nu
\rangle, $$
where $\lambda,\mu,\nu$ are partitions of $n$, $\chi^\rho$ is the
irreducible character of the symmetric group $S_n$ indexed by $\rho$,
and the scalar product is the standard one on class functions on
$S_n$.

isa polynomial of the $x_i$ and $y_j$ together. I think you're trying to ask something about subrings, but it isn't clear. $\endgroup$ – user44191 Mar 12 at 16:55