10
$\begingroup$

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.

$\endgroup$
1
9
$\begingroup$

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 $n^2!/n!$.

$\endgroup$
2
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.