Computing weight $q$-multiplicities for the representations of the simple Lie algebras

The multiplicity of a weight $\mu$ in an irreducible representation of a simple Lie algebra $\mathfrak{g}$ with highest weight $\lambda$ can be computed via the use of Kostant’s weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a $q$-analog of Kostant’s weight multiplicity and present a SageMath program to compute $q$-multiplicities for the simple Lie algebras.