Multiplicity function for tensor powers of modules of the $A_n$ algebra

We consider the decomposition of the $p$th tensor power of the module $L^{\omega_1}$ over the algebra $A_n$ into irreducible modules, $(L^{\omega_1})^{\otimes p}=\sum_{\nu}m(\nu,p)L^{\nu}$. This problem occurs, for example, in finding the spectrum of an invariant Hamiltonian of a spin chain with $p$ nodes. To solve the problem, we propose using the Weyl symmetry properties. For constructing the coefficients $m(\nu,p)$ as functions of $p$, we develop an algorithm applicable to powers of an arbitrary module. We explicitly write an expression for the multiplicities $m(\nu,p)$ in the decomposition of powers of the first fundamental module of $sl(n+1)$. Based on the obtained results, we find new properties of systems of orthogonal polynomials (multivariate Chebyshev polynomials). Our algorithm can also be applied to tensor powers of modules of other simple Lie algebras.