Some spherical function values for two-row tableaux and Young subgroups with three factors

Abstract

A Young subgroup of the symmetric group ${\mathcal{S}}_N$ with three factors, is realized as the stabilizer $G_n$ of a monomial $x^{\hat{I}}»$ ( $=x_1^{\hat{I}{»}_1}{x}_2^{\hat{I}{»}_2}\cdots x_N^{\hat{I}{»}_N}$) with $\hat{I}»=\left(d_1^{n_1},d_2^{n_2},d_3^{n_3}\right)$ (meaning $d_j$ is repeated $n_j$ times, $1\le j\le 3$), thus is isomorphic to the direct product ${\mathcal{S}}_{n_1}\times {\mathcal{S}}_{n_2}\times {\mathcal{S}}_{n_3}$. The orbit of $x^{\hat{I}}»$ under the action of ${\mathcal{S}}_N$ (by permutation of coordinates) spans a module $V_{\hat{I}}»$, the representation induced from the identity representation of $G_n$. The space $V_{\hat{I}}»$ decomposes into a direct sum of irreducible $\mathcal{S}$-modules. The spherical function is defined for each of these, it is the character of the module averaged over the group $G_n$. This paper concerns the value of certain spherical functions evaluated at a cycle which has no more than one entry in each of the three intervals $I_j=\left\{i:\hat{I}{»}_i=d_j\right\},1\le j\le 3$. These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by V. Gorin and the author (arXiv:2412:01938v1). The present paper determines the spherical function values for ${\mathcal{S}}_N$-modules $V$ of two-row tableau type, corresponding to Young tableaux of shape $\left[N-k,k\right]$. The method is based on analyzing the effect of a cycle on $G_n$-invariant elements of $V$. These are constructed in terms of Hahn polynomials in two variables.

14 pages. arXiv admin note: text overlap with arXiv:2503.04547