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$\DeclareMathOperator\U{U}\DeclareMathOperator\O{O}$Schur functions $s_\lambda(x)$ with $\lambda\vdash n$ are simultaneously the irreducible characters of the unitary group $\U(N)$ and proportional to Jack polynomials $J_\lambda^{\alpha}(x)$ with parameter $\alpha=1$, $J_\lambda^1(x)=\frac{n!}{\chi_\lambda(1)}s_\lambda(x)$, where $\chi_\lambda$ are the irreducible characters of the permutation group $S_n$. As a function of $N$, $p_\lambda^{\U}(N)=J_\lambda^1(1^N)$ is a monic polynomial.

Two important relatives of the Schur functions are the irreducible characters of the orthogonal group $\O(N)$, call them $o_\lambda(x)$, and the zonal polynomials $Z_\lambda(x)$, which are Jack polynomials with parameter $\alpha=2$. Both $p_\lambda^{\O}(N)=\frac{n!}{\chi_\lambda(1)}o_\lambda(1^N)$ and $p_\lambda^Z(N)=Z_\lambda(1^N)$ are monic polynomials in $N$.

I have been led to conjecture the following relation between the reciprocals of these polynomials: $$ \sum_{\lambda \vdash n}\frac{\chi_{2\lambda}(1)G_{\lambda\gamma}}{p_\lambda^Z(N)}=\frac{(2n)!}{2^nn!}\frac{\chi_\gamma(1)}{p_\gamma^{\O}(N)},$$ where $2\lambda=(2\lambda_1,2\lambda_2,\dotsc)$ and $$ G_{\lambda\gamma}=\sum_{\mu\vdash n}C_\mu \omega_\lambda(\mu)\chi_\gamma(\mu).$$ Here $C_\mu$ is the size of the conjugacy class in $S_n$ of elements with cycle type $\mu$, and $\omega_\lambda(\mu)$ are zonal spherical functions of $S_{2n}$ with respect to the hyperoctahedral group.

This conjectured relation appeared in connection with immanants of random elements from $\O(N)$ (Oliveira and Novaes - On the immanants of blocks from random matrices in some unitary ensembles) and also with commutators of random elements from $\O(N)$ (Palheta, Barbosa, and Novaes - Commutators of random matrices from the unitary and orthogonal groups).

It is easy to prove that this relation holds for large $N$, but it seems to be true for every finite $N$.

Is this relation known? If not, any idea how to prove it? (Something similar holds for the symplectic analogues, but I omit it for simplicity.)

EDIT:

I don't know if this sheds any light into the conjecture, but, since $s_\gamma=\frac{1}{n}\sum_\mu C_\mu\chi_\gamma(\mu)p_\mu$ and $p_\mu=\frac{2^nn!}{(2n)!}\sum_\lambda \chi_{2\lambda}(1)\omega_\lambda(\mu)Z_\lambda$, it follows that the quantities $\chi_{2\lambda}(1)G_{\lambda\gamma}$ are in fact the coefficients in the expansion of Schur functions into Zonal polynomials, $s_\gamma=\frac{2^n}{(2n)!}\sum_\lambda \chi_{2\lambda}(1)G_{\lambda\gamma}Z_\lambda$.

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    $\begingroup$ That's an interesting question! Note, the Schur polynomials evaluated at $1^N$ gives an Ehrhart polynomial of a certain polytope. I wonder if there are similar interpretations for the Zonal functions... $\endgroup$ Apr 29 '20 at 13:27
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This conjecture has now been proved by Valentin Bonzom, Guillaume Chapuy and Maciej Dołęga in their paper $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and O(N)-BGW integral.

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