For so4, so5, sp4, and g2, the paper lists linear inequalities that identify exactly which Weyl group elements contribute to Kostant's weight multiplicity formula for any dominant weight.
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Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras
For so4, so5, sp4, and g2, the paper lists linear inequalities that identify exactly which Weyl group elements contribute to Kostant's weight multiplicity formula for any dominant weight.