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

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 $so(5)$.