Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Author: Dunkl, Charles F.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/213523
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups. Charles F. Dunkl. SIGMA 21 (2025), 053, 17 pages

A Young subgroup of the symmetric group N, the permutation group of {1, 2, …, }, is generated by a subset of the adjacent transpositions {(, +1)∣1 ≤ < }. Such a group is realized as the stabilizer ₙ of a monomial λ (=λ¹₁λ²₂ ⋯ λᴺN) with λ = (ⁿ¹₁, ⁿ²₂, …, ⁿᵖₚ) (meaning ⱼ is repeated ⱼ times, 1 ≤ ≤ , and ₁ > ₂ > ⋯ > ₚ ≥ 0), thus it is isomorphic to the direct product ₙ₁ × ₙ₂ × ⋯ ×ₙₚ. The interval {1, 2, …, } is a union of disjoint sets ⱼ = { ∣ λᵢ = ⱼ}. The orbit of λ under the action of N (by permutation of coordinates) spans a module λ, the representation induced from the identity representation of ₙ. The space λ decomposes into a direct sum of irreducible N-modules. The spherical function is defined for each of these; it is the character of the module averaged over the group ₙ. This paper concerns the value of certain spherical functions evaluated at a cycle that has no more than one entry in each interval ⱼ. These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by . Gorin and the author [arXiv:2412:01938]. In particular, the present paper determines the spherical function value for N-modules of hook tableau type, corresponding to Young tableaux of shape [ − , 1ᵇ].