Let $M_n$ be the linear space of $n\times n$ matrices. The product of symmetric groups $S_n\times S_n$ acts naturally on $M_n$, and thus induces an action on the coordinate algebra $k[M_n]$. Is there any research on the invariant algebra $k[M_n]^{S_n\times S_n}$ in the literature? Does it have a name so that I may google it? I am also interested in the invariant algebra for the diagonal action of $S_n$. Thank you.
Here is a way to obtain some information on the invariant algebra. Let $G$ denote the permutation group isomorphic to $S_n\times S_n$ acting on an $n\times n$ array by permuting rows and columns. If $w\in G$, then let $c_i(w)$ denote the number of cycles of length $i$ in $w$. Let $f(r)$ denote the dimension of the degree $r$ part of $k[M_n]^{S_n\times S_n}$. By Molien's theorem (e.g., https://en.wikipedia.org/wiki/Molien_series), we have $$ \sum_{r\geq 0}f(r)x^r = \frac{1}{n!^2} \sum_{w\in G}\frac{1}{\prod_{i\geq 1}(1x^i)^{c_i(w)}}. $$

$\begingroup$ Yes, it is a good point to start. I'd like to thank Prof. Stanley for confirming that there is no enough research on this topic. $\endgroup$ – Jiarui Fei Aug 14 '16 at 22:38