0
$\begingroup$

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 $s_{ij}=x_i+y_j$?

$\endgroup$
  • 1
    $\begingroup$ 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 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 at 15:02
  • 1
    $\begingroup$ 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 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 at 17:53
0
$\begingroup$

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 $S_n$.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.