# A Closed Character Formula for Symmetric Powers of Irreducible Representations

Abstract : 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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https://hal.inria.fr/hal-01186237
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### Citation

Stavros Kousidis. A Closed Character Formula for Symmetric Powers of Irreducible Representations. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.833-844, ⟨10.46298/dmtcs.2811⟩. ⟨hal-01186237⟩

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