A $n$-variable function $f$ is a symmetric function if $$f(x_1,x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})$$ for every permutation $\sigma \in S_n$. In particular, if $f$ is a polynomial, then $f$ is a symmetric polynomial. These objects have been studied extensively.

I wonder if the following generalization has been studied. A $n^2$-variable function $f$ is an $S_n$-symmetric function if $$f(x_{11}, x_{12}, \ldots, x_{1n}, \ldots, x_{nn}) = f(x_{\sigma(1)\sigma(1)}, x_{\sigma(1)\sigma(2)}, \ldots, x_{\sigma(1)\sigma(n)}, \ldots, x_{\sigma(n)\sigma(n)})$$ for every permutation $\sigma \in S_n$.

I think such objects must have been studied as they are so natural. But I don't know the keywords and couldn't find the literature.

I'm very grateful if anyone could provide information on them. Thanks in advance.


Let $w\in S_n$ (the symmetric group) have cycle type $\lambda =(\lambda_1,\dots, \lambda_\ell)\vdash n$, where $\ell=\ell(\lambda)$ is the length (number of nonzero parts) of $\lambda$. Then the induced action of $w$ on $[n]\times [n]$ (where $[n]=\{1,2,\dots,n\}$) has cycle enumerator $$ \prod_{i=1}^{\ell(\lambda)} \prod_{j=1}^{\ell(\lambda)} z_{\mathrm{lcm}(\lambda_i,\lambda_j)}^{\mathrm{gcd}(\lambda_i,\lambda_j)}. $$ Let $f_n(d)$ be the dimension of the space of complex polynomials in the variables $x_{ij}$, $1\leq i,j\leq n$, that are homogeneous of degree $d$ and $G$-invariant. Then by Molien's theorem, $$ F_n(x):=\sum_{d\geq 0} f_n(d)x^d $$ $$ \ \ = \frac{1}{n!}\sum_{\lambda\vdash n} \frac{n!}{z_\lambda}\prod_{i=1}^{\ell(\lambda)} \prod_{j=1}^{\ell(\lambda)} \frac{1}{\left(1-x^{\mathrm{lcm}(\lambda_i,\lambda_j)}\right)^ {\mathrm{gcd}(\lambda_i,\lambda_j)}}. $$ I am using standard symmetric function notation, so $n!/z_\lambda$ is the number of permutations in $S_n$ of cycle type $\lambda$. For instance, $$ F_1(x) = \frac{1}{1-x} $$ $$ F_2(x) = \frac{1+x^2}{(1-x)^4(1+x)^2} $$ $$ F_3(x) = \frac{1+3x^2+10x^3+16x^4+12x^5+16x^6+10x^7+3x^8+x^{10}} {(1-x)^9(1+x)^4(1+x+x^2)^3}. $$ Addendum. The invariant theory of finite groups, such as can be found here, can be used to obtain some further information about the ring $R$ of invariant polynomials. For instance, if $S$ is the subring of all symmetric functions in the $x_{ij}$'s, then $R$ is a finitely-generated free $S$-module of rank $n^2!/n!$.

  • $\begingroup$ I am confused. I think $f_n(1)$ should be $2$. The sum of diagonal elements and the sum of off-diagonal elements form a basis. But if $F_1(x) = \frac{1}{1-x}$, then $f_n(1) = 1$. I am sorry that the origonal notation "$S_n \times S_n$-invariant" is misleading. $\endgroup$
    – MMM
    Oct 16 '20 at 6:13
  • 3
    $\begingroup$ @MMM $f_n(1)$ is not the coefficient of $x^n$ in $F_1(x)$, it is the coefficient of $x$ in $F_n(x)$. It is easy to see that for $F_2(x)$ and $F_3(x)$ as above, the coefficient of $x$ is indeed equal to $2$. $\endgroup$ Oct 16 '20 at 6:46

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