The permutation group $S\_{2n}$ has $H\_{2,n}=S\_2\wr S\_n$ as a subgroup. The plethysm $h\_n(h\_2)=\sum\_{\lambda\vdash n}s\_{2\lambda}$ is well known.


The zonal spherical functions $\omega\_\lambda(g)=\frac{1}{H\_{2,n}}\sum\_{h\in H\_{2,n}}\chi\_{2\lambda}(gh)$ are responsible for the transition between power sum symmetric polynomials and zonal polynomials, $$Z\_\mu \overset{\omega\_\lambda(\mu)} \longrightarrow p\_\lambda.$$


My question is what happens when we change the subgroup to $H\_{n,2}=S\_n\wr S\_2$. The plethysm $h\_2(h\_n)=\sum\_{(m\_1,m\_2)\vdash n}s\_{(2m\_1,2m\_2)}$ is also well known. Are the corresponding zonal spherical functions known? Are they responsible for the transition between already studied clases of symmetric polynomials?



Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials
The corresponding orthogonal polynomials should be closely related to the *Eberlein polynomials*, i.e., their coefficients should be closely related to the zonal spherical functions of the Gelfand pair $(S\_{2n},S\_{n} \times S\_n)$, see VII.1 Ex. 13 of Macdonald's text, for example.


In particular, the zonal spherical functions of $(S\_{2n},S\_{n} \wr S\_2)$ are eigenvectors of the so-called *folded Johnson graphs*, a family of graphs derived from the Johnson scheme $\mathcal{J}(2n,n)$ by identifying antipodal vertices, i.e., identifying each $n$-set with its complement. It should be straightforward to arrive at an explicit expression for the zonal spherical functions of $(S\_{2n},S\_{n} \wr S\_2)$ from the known expression of the zonal spherical functions of $(S\_{2n},S\_{n} \times S\_n)$, which are eigenvectors of the Johnson graphs. There's probably a slicker way of showing this that avoids this detour through algebraic graph theory that I'm suggesting.


