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

A closed character formula for symmetric powers of irreducible representations