Generalized Pascal's triangles and singular elements of modules of Lie algebras.
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| Title: | Generalized Pascal's triangles and singular elements of modules of Lie algebras. |
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| Authors: | Lyakhovsky, V.1 lyakvladimir@yandex.ru, Postnova, O.1 postnova.olga@gmail.com |
| Source: | Theoretical & Mathematical Physics. Oct2015, Vol. 185 Issue 1, p1481-1491. 11p. |
| Subjects: | Pascal's triangle, Mathematical singularities, Generalization, Modules (Algebra), Lie algebras, Multiplicity (Mathematics) |
| Abstract: | We consider the problem of determining the multiplicity function $$m_\xi ^{{ \otimes ^p}\omega }$$ in the tensor power decomposition of a module of a semisimple algebra g into irreducible submodules. For this, we propose to pass to the corresponding decomposition of a singular element Ψ((L )) of the module tensor power into singular elements of irreducible submodules and formulate the problem of determining the function $$M_\xi ^{{ \otimes ^p}\omega }$$. This function satisfies a system of recurrence relations that corresponds to the procedure for multiplying modules. To solve this problem, we introduce a special combinatorial object, a generalized (g,ω) pyramid, i.e., a set of numbers ( p, { mi}) satisfying the same system of recurrence relations. We prove that $$M_\xi ^{{ \otimes ^p}\omega }$$ can be represented as a linear combination of the corresponding ( p, { mi}). We illustrate the obtained solution with several examples of modules of the algebras sl(3) and so(5). [ABSTRACT FROM AUTHOR] |
| Copyright of Theoretical & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: Generalized Pascal's triangles and singular elements of modules of Lie algebras. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Lyakhovsky%2C+V%2E%22">Lyakhovsky, V.</searchLink><relatesTo>1</relatesTo><i> lyakvladimir@yandex.ru</i><br /><searchLink fieldCode="AR" term="%22Postnova%2C+O%2E%22">Postnova, O.</searchLink><relatesTo>1</relatesTo><i> postnova.olga@gmail.com</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Theoretical+%26+Mathematical+Physics%22">Theoretical & Mathematical Physics</searchLink>. Oct2015, Vol. 185 Issue 1, p1481-1491. 11p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Pascal's+triangle%22">Pascal's triangle</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+singularities%22">Mathematical singularities</searchLink><br /><searchLink fieldCode="DE" term="%22Generalization%22">Generalization</searchLink><br /><searchLink fieldCode="DE" term="%22Modules+%28Algebra%29%22">Modules (Algebra)</searchLink><br /><searchLink fieldCode="DE" term="%22Lie+algebras%22">Lie algebras</searchLink><br /><searchLink fieldCode="DE" term="%22Multiplicity+%28Mathematics%29%22">Multiplicity (Mathematics)</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We consider the problem of determining the multiplicity function $$m_\xi ^{{ \otimes ^p}\omega }$$ in the tensor power decomposition of a module of a semisimple algebra g into irreducible submodules. For this, we propose to pass to the corresponding decomposition of a singular element Ψ((L )) of the module tensor power into singular elements of irreducible submodules and formulate the problem of determining the function $$M_\xi ^{{ \otimes ^p}\omega }$$. This function satisfies a system of recurrence relations that corresponds to the procedure for multiplying modules. To solve this problem, we introduce a special combinatorial object, a generalized (g,ω) pyramid, i.e., a set of numbers ( p, { mi}) satisfying the same system of recurrence relations. We prove that $$M_\xi ^{{ \otimes ^p}\omega }$$ can be represented as a linear combination of the corresponding ( p, { mi}). We illustrate the obtained solution with several examples of modules of the algebras sl(3) and so(5). [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Theoretical & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s11232-015-0357-0 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 11 StartPage: 1481 Subjects: – SubjectFull: Pascal's triangle Type: general – SubjectFull: Mathematical singularities Type: general – SubjectFull: Generalization Type: general – SubjectFull: Modules (Algebra) Type: general – SubjectFull: Lie algebras Type: general – SubjectFull: Multiplicity (Mathematics) Type: general Titles: – TitleFull: Generalized Pascal's triangles and singular elements of modules of Lie algebras. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Lyakhovsky, V. – PersonEntity: Name: NameFull: Postnova, O. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 10 Text: Oct2015 Type: published Y: 2015 Identifiers: – Type: issn-print Value: 00405779 Numbering: – Type: volume Value: 185 – Type: issue Value: 1 Titles: – TitleFull: Theoretical & Mathematical Physics Type: main |
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