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More Character Formulas

  • William Fulton
  • Joe Harris
Part of the Graduate Texts in Mathematics book series (GTM, volume 129)

Abstract

In this lecture we give two more formulas for the multiplicities of an irreducible representation of a semisimple Lie algebra or group. First, Freudenthal’s formula 025.1) gives a straightforward way of calculating the multiplicity of a given weight once we know the multiplicity of all higher ones. This in turn allows us to prove in §25.2 the Weyl character formula, as well as another multiplicity formula due to Kostant. Finally, in §25.3 we give Steinberg’s formula for the decomposition of the tensor product of two arbitrary irreducible representations of a semisimple Lie algebra, and also give formulas for some pairs \( \mathfrak{h} \subset \mathfrak{g}\) 11 c g for the decomposition of the restriction to \( \mathfrak{h}\) of irreducible representations of \(\mathfrak{g} \).

Keywords

Tensor Product Irreducible Representation Weyl Group Weight Space Casimir Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • William Fulton
    • 1
  • Joe Harris
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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