Spectral Theory of the Nazarov-Sklyanin Lax Operator
In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator : [] → [] where is the ring of symmetric functions, and w is a variable. In this paper, we (1) establish a cyclic decomposition [] ≅ ⨁λ (λ, ) into finite-dimensional -cyclic subspaces in which Ja...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212021 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Spectral Theory of the Nazarov-Sklyanin Lax Operator. Ryan Mickler and Alexander Moll. SIGMA 19 (2023), 063, 22 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862620219348877312 |
|---|---|
| author | Mickler, Ryan Moll, Alexander |
| author_facet | Mickler, Ryan Moll, Alexander |
| citation_txt | Spectral Theory of the Nazarov-Sklyanin Lax Operator. Ryan Mickler and Alexander Moll. SIGMA 19 (2023), 063, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator : [] → [] where is the ring of symmetric functions, and w is a variable. In this paper, we (1) establish a cyclic decomposition [] ≅ ⨁λ (λ, ) into finite-dimensional -cyclic subspaces in which Jack polynomials λ may be taken as cyclic vectors and (2) prove that the restriction of to each (jλ, ) has simple spectrum given by the anisotropic contents [] of the addable corners of the Young diagram of λ. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated with , both established by Nazarov-Sklyanin. Finally, we conjecture that the -eigenfunctions ˢλ ∈ [] {with eigenvalue [] and constant term} ˢλ|₌₀ = λ are polynomials in the rescaled power sum basis μˡ of [] with integer coefficients.
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| first_indexed | 2026-03-14T12:36:10Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212021 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T12:36:10Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mickler, Ryan Moll, Alexander 2026-01-22T09:21:20Z 2023 Spectral Theory of the Nazarov-Sklyanin Lax Operator. Ryan Mickler and Alexander Moll. SIGMA 19 (2023), 063, 22 pages 1815-0659 2020 Mathematics Subject Classification: 05E05; 33D52; 37K10; 47B35 arXiv:2211.01586 https://nasplib.isofts.kiev.ua/handle/123456789/212021 https://doi.org/10.3842/SIGMA.2023.063 In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator : [] → [] where is the ring of symmetric functions, and w is a variable. In this paper, we (1) establish a cyclic decomposition [] ≅ ⨁λ (λ, ) into finite-dimensional -cyclic subspaces in which Jack polynomials λ may be taken as cyclic vectors and (2) prove that the restriction of to each (jλ, ) has simple spectrum given by the anisotropic contents [] of the addable corners of the Young diagram of λ. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated with , both established by Nazarov-Sklyanin. Finally, we conjecture that the -eigenfunctions ˢλ ∈ [] {with eigenvalue [] and constant term} ˢλ|₌₀ = λ are polynomials in the rescaled power sum basis μˡ of [] with integer coefficients. The authors would like to thank the referees for many helpful comments and suggestions. We would also like to express our sincere thanks to the staff at Darwin’s Ltd. coffee and sandwich shop on Cambridge Street in Cambridge, MA, for supporting our collaboration during the years 2014–2019. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spectral Theory of the Nazarov-Sklyanin Lax Operator Article published earlier |
| spellingShingle | Spectral Theory of the Nazarov-Sklyanin Lax Operator Mickler, Ryan Moll, Alexander |
| title | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_full | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_fullStr | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_full_unstemmed | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_short | Spectral Theory of the Nazarov-Sklyanin Lax Operator |
| title_sort | spectral theory of the nazarov-sklyanin lax operator |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212021 |
| work_keys_str_mv | AT micklerryan spectraltheoryofthenazarovsklyaninlaxoperator AT mollalexander spectraltheoryofthenazarovsklyaninlaxoperator |