More Character Formulas

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 \(\mathfrak{g} \).