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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 $s_{ij}=x_i+y_j$?

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    $\begingroup$ 1) Is $f$ invariant under permutation that mix the $x_i$ and $y_j$? 2) What do you mean by "Expanded in terms of polynomials of $x_i$ and $y_j$ together"? 3) In your "for instance", say $m = n = 1$. Then are you asking whether $xy$ can be written in terms of $x + y$? $\endgroup$
    – user44191
    Mar 12 '19 at 14:51
  • $\begingroup$ 1) no, $f$ is not given to be invariant under anything that mixes $x_i$ and $y_j$. 2) if we take any symmetric function in $mn$ variables and replace them with the variables $x_iy_j$, for instance, we get such an $f$. This $f$ can be expressed as elementary functions in the $x_iy_j$, and I would like to characterize this property. $\endgroup$
    – user133644
    Mar 12 '19 at 15:02
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    $\begingroup$ I still don't understand the property you're looking for. Any $f$ can be expanded in termss of polynomials of the $x_i$ and $y_j$ together, because any $f$ is a 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 '19 at 16:55
  • $\begingroup$ Suppose $S$ is the ring of symmetric polynomials in $mn$ variables $s_{ij}$. Then there is a map from $S$ to the polynomials which are symmetric in $x_i$ and $y_j$ separately, by sending $s_{ij}$ to $x_iy_j$. What is the image of this map? What about the cokernel? Similar questions if you replace $x_iy_j$ with some symmetric function of $x_i$ and $y_j$. $\endgroup$
    – user133644
    Mar 12 '19 at 17:53
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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 $S_n$.

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