Generalized Pascal's triangles and singular elements of modules of Lie algebras.

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Bibliographic Details
Title:	Generalized Pascal's triangles and singular elements of modules of Lie algebras.
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]
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Database:	Engineering Source

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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]
ISSN:	00405779
DOI:	10.1007/s11232-015-0357-0