# Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials

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?

• How do you define the zonal polynomial when $h_n[h_k]$ is not multiplicity-free? Mar 26, 2021 at 16:16
• @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. Mar 26, 2021 at 17:09
• @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. Mar 26, 2021 at 17:10

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.
• 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. Mar 28, 2021 at 23:36