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Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups
Charles F. Dunkl Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA
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 adjacent
transpositions
$\{ ( i,i+1) \mid 1\leq i<N\}$. Such a group is realized as the stabilizer
$G_{n}$ of a monomial \smash{
$x^{\lambda}$ $\big({=}\,x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots x_{N}^{\lambda_{N}}\big)$} with \smash{${\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
$\bigl[ N-b,1^{b}\bigr]$.
Keywords:
spherical functions, subgroups of the symmetric group, hook tableaux, alternating polynomials.
MSC: 20C30,
43A90,
20B30 Received: March 12, 2025; in final form
June 30, 2025; Published online
July 8, 2025
Language: English
DOI:
10.3842/SIGMA.2025.053
Fulltext:
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