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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?

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  • $\begingroup$ How do you define the zonal polynomial when $h_n[h_k]$ is not multiplicity-free? $\endgroup$ – Richard Stanley Mar 26 at 16:16
  • $\begingroup$ @RichardStanley Actually the argument of $\omega$ are the double cosets $H\G/H$, which can be represented by equivalence classes of $n\times n$ matrices of non-negative integers entries that sum to $k$. So the first question would be if there are symmetric polynomials associated with that. $\endgroup$ – thedude Mar 26 at 17:09
  • $\begingroup$ @RichardStanley I removed the last bit of the question, lest it introduce confusion. In any case, $h_2[h_n]$ is multiplicity-free. In this case I think the "zonal polynomials" could be defined. $\endgroup$ – thedude Mar 26 at 17:10
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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.

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  • $\begingroup$ This seems very interesting. Could you point to some other references besides McDonald? $\endgroup$ – thedude Mar 28 at 21:07
  • $\begingroup$ If it is the connection to association schemes that seems interesting, then look at Godsil and Meagher's text on Erdos-Ko-Rado combinatorics and Bannai and Ito's text on association schemes. The latter spells out the connection between association schemes and Gelfand pairs a bit more. I don't think you'll find a nice formula for the spherical functions of $(S_{2n},S_n \wr S_2)$ in print. This gap in the literature is likely due to its closeness to $(S_{2n},S_n \times S_n)$. Finally, there's probably a $q$-analogue of $(S_{2n},S_n \wr S_2)$ that has a name in the finite geometry literature. $\endgroup$ – Nathan Lindzey Mar 28 at 23:36

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