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| Резюме: | 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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| ISSN: | 1815-0659 |