Stavros Kousidis

A Closed Character Formula for Symmetric Powers of Irreducible Representations
dmtcs:2811 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

https://doi.org/10.46298/dmtcs.2811
A Closed Character Formula for Symmetric Powers of Irreducible RepresentationsArticle
Authors: Stavros Kousidis ^{1}
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Stavros Kousidis
1 Max Planck Institute for Mathematics
We prove a closed character formula for the symmetric powers $S^N V(λ )$ of a fixed irreducible representation $V(λ )$ of a complex semisimple 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 residuetype 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.
Letao Zhang, 2015, Character formulas on cohomology of deformations of Hilbert schemes ofK3 surfaces, arXiv (Cornell University), 92, 3, pp. 675688, 10.1112/jlms/jdv041, https://arxiv.org/abs/1301.6129.