I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize.
For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$ (only integer dimensions, not fractals :) and $\bar{X}$ are vectors? Is there a way to systematically generate symmetric functions, $f_1, f_2, ..., f_N$ so that $f_i(\bar{X}_1, ..., \bar{X}_N)$ generates a vector of dimension $D$?
If this is not possible in general, can we accomplish it for the limited case when $N=2$ and $D>1$?