If I have $N$ variables, $a_i$, $i=1,...,N$, then any symmetric polynomial in these can be expressed as a polynomial in the elementary symmetric polynomials:
$$ e_k = \sum_{i_1<...<i_k} a_{i_1} ...a_{i_k}, \;\;\; k=1,...,N $$
Now suppose I instead have $N$ functions, $f_i(x)$, $i=1,...,N$. Now I'm interested in symmetric polynomials in the functions, which may be evaluated at different arguments, eg, something like (for $N=2$):
$$ {f_1(x)}^2 f_2(y) + {f_2(x)}^2 f_1(y) $$
Is there some analogous set of ``generating functions'' for these symmetric polynomials? Perhaps it is natural to define
$$ e_k(x_1,...x_k) = f_{1}(x_1) ...f_{k}(x_k) + \text{permutations}, \;\;\; k=1,...,N $$ which I may take arbitrary polynomials in and evaluate at arbitrary arguments. But I'm not sure this is enough to capture everything.