A closed character formula for symmetric powers of irreducible representations

We prove a closed character formula for the symmetric powers $S^N V(λ)$ of a fixed irreducible representation $V(λ)$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(λ)$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(λ)$ and explain the connections between our character formula, vector partition functions and iterated partial fraction decomposition.

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