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edited Sep 3, 2014 at 20:51

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I suspect that the answer you seek can be deduced from the Kostant Multiplicity Formula. It gives the multiplicity of a weight in an irreducible complex representation of a compact Lie group $G$. One reference would be: A Formula for the Multiplicity of a Weight, by Kostant.

answered Sep 3, 2014 at 17:32

Peter Crooks

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