Geodesic
Generalized Pascal's triangles and singular elements of modules of Lie algebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 1, pp. 139-150 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\mathfrak{g}$ into irreducible submodules. For this, we propose to pass to the corresponding decomposition of a singular element $\Psi((L_g^\omega)^{\otimes^p})$ 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 $(g,\omega)$ pyramid, i.e., a set of numbers $(p,\{m_i\})_{g,\omega}$ 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 $(p,\{m_i\})_{g,\omega}$. We illustrate the obtained solution with several examples of modules of the algebras $sl(3)$ and $so(5)$.
Keywords: theory of Lie algebra representation, tensor product of modules, Weyl formula. 
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     title = {Generalized {Pascal's} triangles and singular elements of modules of {Lie} algebras},
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V. D. Lyakhovsky; O. V. Postnova. Generalized Pascal's triangles and singular elements of modules of Lie algebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 1, pp. 139-150. http://geodesic.mathdoc.fr/item/TMF_2015_185_1_a12/

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