TITLE: Hyper-symmetric polynomials (reference request)
QUESTION [9 upvotes]: 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.

REPLY [4 votes]: 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
  $$ \sum_{r\geq 0}f(r)x^r = \frac{1}{n!^2}
   \sum_{w\in G}\frac{1}{\prod_{i\geq
  1}(1-x^i)^{c_i(w)}}. $$