An algorithm for computing weight multiplicities in irreducible modules for complex semisimple Lie algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ having rank $l$ and let $V=L(\lambda)$ be an irreducible finite-dimensional $\mathfrak{g}$-module having highest weight $\lambda.$ Computations of weight multiplicities in $V,$ usually based on Freudenthal's formula, are in general difficult to carry out in large ranks or for $\lambda$ with large coefficients (in terms of the fundamental weights). In this paper, we first show that in some situations, these coefficients can be "lowered" in order to simplify the calculations. We then investigate how this can be used to improve the aforementioned formula of Freudenthal, leading to a more efficient version of the latter in terms of complexity as well as a way of dealing with certain computations in unbounded ranks. We conclude by illustrating the last assertion with a concrete example.