From Jack to Double Jack Polynomials via the Supersymmetric Bridge
The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superp...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2015 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2015
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147121 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | From Jack to Double Jack Polynomials via the Supersymmetric Bridge / L. Lapointe, P. Mathieu // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 46 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-147121 |
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Lapointe, L. Mathieu, P. 2019-02-13T17:00:55Z 2019-02-13T17:00:55Z 2015 From Jack to Double Jack Polynomials via the Supersymmetric Bridge / L. Lapointe, P. Mathieu // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 46 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E05; 81Q60; 81R12; 37J35 DOI:10.3842/SIGMA.2015.051 https://nasplib.isofts.kiev.ua/handle/123456789/147121 The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine slˆ₂ algebra. This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html. We thank Olivier Blondeau-Fournier for his collaboration on [9]. This work was supported by the Natural Sciences and Engineering Research Council of Canada; the Fondo Nacional de Desarrollo Cient´ıfico y Tecnol´ogico de Chile grant #1130696. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications From Jack to Double Jack Polynomials via the Supersymmetric Bridge Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
From Jack to Double Jack Polynomials via the Supersymmetric Bridge |
| spellingShingle |
From Jack to Double Jack Polynomials via the Supersymmetric Bridge Lapointe, L. Mathieu, P. |
| title_short |
From Jack to Double Jack Polynomials via the Supersymmetric Bridge |
| title_full |
From Jack to Double Jack Polynomials via the Supersymmetric Bridge |
| title_fullStr |
From Jack to Double Jack Polynomials via the Supersymmetric Bridge |
| title_full_unstemmed |
From Jack to Double Jack Polynomials via the Supersymmetric Bridge |
| title_sort |
from jack to double jack polynomials via the supersymmetric bridge |
| author |
Lapointe, L. Mathieu, P. |
| author_facet |
Lapointe, L. Mathieu, P. |
| publishDate |
2015 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine slˆ₂ algebra.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147121 |
| citation_txt |
From Jack to Double Jack Polynomials via the Supersymmetric Bridge / L. Lapointe, P. Mathieu // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 46 назв. — англ. |
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2025-12-07T15:35:23Z |
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2025-12-07T15:35:23Z |
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