Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups

Abstract

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 $\{(i,i+1)\mid 1\le i<N\}$. Such a group 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 $\hat{I}»=(d_1^{n_1},d_2^{n_2},\dots ,d_p^{n_p})$ (meaning $d_j$ is repeated $n_j$ times, $1\le j\le p$, and $d_1>d_2>\cdots >d_p\ge 0$), 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 $I_j=\{i\mid \hat{I}{»}_i=d_j\}$. 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 ${\mathcal{S}}_N$-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 $[N-b,1^b]$.