Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One
We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one...
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| Дата: | 2014 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2014
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146404 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1464042019-02-10T01:23:33Z Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One Maarten van Pruijssen Román, P. We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change Ψ₀. We analyze the base change and derive several properties. The most important one is that Ψ₀ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for Ψ0. The weight W is also determined by Ψ₀. We provide an algorithm to calculate Ψ₀ explicitly. For the pair (USp(2n),USp(2n−2)×USp(2)) we have implemented the algorithm in GAP so that individual pairs (W,D) can be calculated explicitly. Finally we classify the Gelfand pairs (G,K) and the K-representations that yield pairs (W,D) of size 2×2 and we provide explicit expressions for most of these cases. 2014 Article Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E46; 33C47 DOI: http://dx.doi.org/10.3842/SIGMA.2014.113 http://dspace.nbuv.gov.ua/handle/123456789/146404 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change Ψ₀. We analyze the base change and derive several properties. The most important one is that Ψ₀ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for Ψ0. The weight W is also determined by Ψ₀. We provide an algorithm to calculate Ψ₀ explicitly. For the pair (USp(2n),USp(2n−2)×USp(2)) we have implemented the algorithm in GAP so that individual pairs (W,D) can be calculated explicitly. Finally we classify the Gelfand pairs (G,K) and the K-representations that yield pairs (W,D) of size 2×2 and we provide explicit expressions for most of these cases. |
| format |
Article |
| author |
Maarten van Pruijssen Román, P. |
| spellingShingle |
Maarten van Pruijssen Román, P. Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Maarten van Pruijssen Román, P. |
| author_sort |
Maarten van Pruijssen |
| title |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_short |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_full |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_fullStr |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_full_unstemmed |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_sort |
matrix valued classical pairs related to compact gelfand pairs of rank one |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146404 |
| citation_txt |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:24:40Z |
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