Branching rules are a systematic way of understanding the multiplicity of irreducible representations in restrictions of representations of Lie groups. In the case of $GL_n$ and orthogonal groups, the branching rules are multiplicity free, but the same is not the case for symplectic groups. The explicit combinatorial description of the multiplicities was given by Lepowsky in his PhD thesis. In 2009, Wallach and Oded showed that this multiplicity corresponds to the dimension of the multiplicity space, which was a representation of $SL_2$ $(=Sp(2))$. In this talk, we give an alternate proof of the same without invoking any partition function machinery. The only assumption for this talk would be the Weyl character formula.

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Last updated: 11 Dec 2023