# Identity involving zonal polynomials and $\operatorname O(N)$ irrep dimensions

$$\DeclareMathOperator\U{U}\DeclareMathOperator\O{O}$$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, $$s_\gamma=\frac{2^n}{(2n)!}\sum_\lambda \chi_{2\lambda}(1)G_{\lambda\gamma}Z_\lambda$$.

• 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... Apr 29, 2020 at 13:27

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.