# Symmetric polynomials in two sets of variables

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 $$s_{ij}=x_i+y_j$$?

• 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 $x + y$? Mar 12 '19 at 14:51
• 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.
– user133644
Mar 12 '19 at 15:02
• 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. Mar 12 '19 at 16:55
• 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 $y_j$.
– user133644
Mar 12 '19 at 17:53

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 $$S_n$$.