Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups
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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/213523 |
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| 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 |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862726599201259520 |
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| author | Dunkl, Charles F. |
| author_facet | Dunkl, Charles F. |
| citation_txt | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups. Charles F. Dunkl. SIGMA 21 (2025), 053, 17 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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].
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| first_indexed | 2026-03-21T09:33:16Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-213523 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T09:33:16Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
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| spelling | Dunkl, Charles F. 2026-02-18T11:24:19Z 2025 Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups. Charles F. Dunkl. SIGMA 21 (2025), 053, 17 pages 1815-0659 2020 Mathematics Subject Classification: 20C30; 43A90; 20B30 arXiv:2503.04547 https://nasplib.isofts.kiev.ua/handle/123456789/213523 https://doi.org/10.3842/SIGMA.2025.053 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]. The author is grateful to the referees whose careful reading and detailed suggestions helped to improve this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups Article published earlier |
| spellingShingle | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups Dunkl, Charles F. |
| title | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_full | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_fullStr | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_full_unstemmed | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_short | Some Spherical Function Values for Hook Tableaux Isotypes and Young Subgroups |
| title_sort | some spherical function values for hook tableaux isotypes and young subgroups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213523 |
| work_keys_str_mv | AT dunklcharlesf somesphericalfunctionvaluesforhooktableauxisotypesandyoungsubgroups |