Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups

A Young subgroup of the symmetric group $\mathcal{S}_{N}$, the permutation group of $\{ 1,2,\dots,N\}$, is generated by a subset of the adjacenttranspositions $\{ ( i,i+1) \mid 1\leq i < N\}$. Such a group is realized as the stabilizer $G_{n}$ of a monomial $x^{\lambda}$ $\big({=}\,x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots x_{N}^{\lambda_{N}}\big)$ with ${\lambda=\bigl( d_{1}^{n_{1}},d_{2}^{n_{2}}, \dots,d_{p}^{n_{p}}\bigr)}$ (meaning $d_{j}$ is repeated $n_{j}$ times, $1\leq j\leq p$, and $d_{1}>d_{2}>\dots>d_{p}\geq0$), thus is isomorphic to the direct product $\mathcal{S}_{n_{1}}\times\mathcal{S}_{n_{2}} \times\cdots\times\mathcal{S}_{n_{p}}$. The interval $\{ 1,2,\dots,N\}$ is a union of disjoint sets $I_{j}= \{ i\mid \lambda_{i}=d_{j} \}$. The orbit of $x^{\lambda}$ under the action of $\mathcal{S}_{N}$ (by permutation of coordinates) spans a module $V_{\lambda}$, the representation induced from the identity representation of $G_{n}$. The space $V_{\lambda}$ decomposes into a direct sum of irreducible $\mathcal{S}_{N}$-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 interval $I_{j}$. These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by V. Gorin and the author [arXiv:2412:01938]. In particular, the present paper determines the spherical function value for $\mathcal{S}_{N}$-modules of hook tableau type, corresponding to Young tableaux of shape $\bigl[ N-b,1^{b}\bigr]$.