Maximally independent polynomial families with row symmetry Introduction:
In the 1-dimensional case, given $m$-variables
$$\mathbf{x} = (x_1,x_2,\dots,x_m)^T,$$
the elementary symmetric polynomials $(e_k(\mathbf{x}))_{k=1}^m$ give a "symmetric basis", in the sense of a maximally independent set / minimally generating set of symmetric polynomials.
I have been trying to generalize such bases to higher dimensions. For example given (real) variables
$$X=\begin{bmatrix}x_1 & y_1 \\ x_2 & y_2 \\ \vdots & \vdots \\ x_m & y_m \end{bmatrix},$$
I am interested in a minimal, generating set of polynomials $f(X)$ whose symmetries are precisely the row permutations. That is, for any $m \times m$ permutation matrix $P$, I want
$$ f(PX)=f(X)$$
to hold. Also any other polynomial with such symmetry should be a function of those in the set.
I found out that I can take the real and imaginary parts of the complexified elementary symmetric polynomials:
$$\left(\Re e_k(x_1+i y_1,x_2+i y_2,\dots, x_m+iy_m) \right)_{k=1}^m,\left(\Im e_k(x_1+i y_1,x_2+i y_2,\dots, x_m+iy_m) \right)_{k=1}^m .$$
My Question(s):
Given $mn$ variables in a rectangular array
$$X=\begin{bmatrix}x_{1,1} & x_{1,2} & \dots & x_{1,n}  \\ x_{2,1} & x_{2,2} & \dots & x_{2,n} \\ \vdots & \vdots & \ddots & \vdots  \\ x_{m,1} & x_{m,2} & \dots & x_{m,n} \end{bmatrix}$$
what are some symmetric bases (symmetries being precisely row permutations of the variables)? Are there any standard ones in the literature? Ones that can be computed efficiently?
My Attempt/Thoughts:
From the 2-dimensional case, I have some hypothesized properties of a basis.

*

*The Jacobian matrix of the basis might have a "generalized Vandermonde" determinant $$\pm \prod_{i<j} \left( (x_{i,1}-x_{j,1})^2 + (x_{i,2}-x_{j,2})^2 + \dots + (x_{i,n}-x_{j,n})^2 \right), $$
vanishing precisely when rows coincide.

*The basis might contain $n$ polynomials of each degree $1\leq d \leq m$, with $mn$ polynomials in total.

*The above gives an underdetermined set of constraints, even with only $2 \times 2$ I have found many polynomials that complete the columns' sums $x_1+x_2, y_1+y_2$ to a basis.

*There might be such families of polynomials in the literature, but I couldn't find them.

Note: Taking the union of the column-wise elementary symmetric polynomials is no good, as there are extra symmetries.
Thank you for any help.
 A: Let $R$ be the ring $k[x_{ij}]$ of polynomials in your $mn$ variables and let $S$ be the subring of polynomials invariant for your $S_n$ action. You are asking for nice elements of $S$.
You say "Also any other polynomial with such symmetry should be a function of those in the set." If this means that any other element of $S$ should be a polynomial function of your chosen set, then that means you want your set $S$ to generate the ring $S$. In this case, you can't get by with $mn$ generators (except for $m=1$ or $n=1$), because $S$ is not a polynomial ring. In the case $m=n=2$ discussed in comments, the Macmahon generators $x_1+x_2$, $y_1+y_2$, $x_1 x_2$, $y_1 y_2$, $x_1 y_1 + x_2 y_2$ generate a maximal ideal $\mathfrak{m}$ of $S$, and they are linearly independent in the Zariski tangent space $\mathfrak{m}/\mathfrak{m}^2$, so you really do need $5$ functions to generate this ring.
If you are willing to accept some sort of weaker notion of "function of", you might be able to say more. For example, if you want any element of $S$ to be a rational function of your polynomials, they you are just asking if $\text{Frac}(S)$ is a purely transcendental field extension of $k$. This is doable, although perhaps uninteresting. Take the $n$ elementary symmetric functions of your first column of variables, and then, for each other column, take the sums $f_{pj}:=\sum_{i=1}^n x_{i1}^p x_{ij}$ for $0 \leq p \leq n-1$ and $2 \leq j \leq m$. Then the elementary symmetric functions determine the $x_{i1}$ up to permutation's and, if the $x_{i1}$'s are distinct, then the $f_{pj}$'s give $n$ linearly independent linear conditions for the $x_{ij}$'s.
