Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 053, 17 pages      arXiv:2503.04547      https://doi.org/10.3842/SIGMA.2025.053

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

Received March 12, 2025, in final form June 30, 2025; Published online July 08, 2025

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 $\{ ( i,i+1) \mid 1\leq i$ < $N\}$. 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 $\bigl[ N-b,1^{b}\bigr]$.

Key words: spherical functions; subgroups of the symmetric group; hook tableaux; alternating polynomials.

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References

1. Ceccherini-Silberstein T., Scarabotti F., Tolli F., Harmonic analysis on finite groups, Cambridge Stud. Adv. Math., Vol. 108, Cambridge University Press, Cambridge, 2008.
2. Chalykh O., Dunkl and Cherednik operators, in Encyclopedia of Mathematical Physics, Vol. 3, Academic Press, Oxford, 2025, 309-327, arXiv:2409.09005.
3. Dunkl C., Gorin V., Eigenvalues of Heckman-Polychronakos operators, arXiv:2412.01938.
4. Heckman G.J., An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), 341-350.
5. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Math. Monogr., The Clarendon Press, New York, 1995.
6. Polychronakos A.P., Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992), 703-705, arXiv:hep-th/9202057.