Visualizing the support of Kostant’s weight multiplicity formula for the rank-2 Lie algebras

The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostant's partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group which contributes nontrivially to the multiplicity of a weight in a highest weight representation of the Lie algebras so₄ (C), so₅ (C), sp₄ (C), and the exceptional Lie algebra g₂. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on sl₃ (C), to provide visualizations of these Weyl alternation sets for all pairs of integral weights and of the Lie algebras considered.