Weyl’s Character Formula for Representations of Semisimple Lie Algebras

Weyl’s character formula is a useful tool in understanding the structure of irreducible representations of semisimple Lie algebras. Even more important than the character formula itself is a corollary, the Weyl dimension formula, which gives a fairly simple expression for the total dimension of the representation in terms of the root system of the Lie algebra. Although the results discussed are valid over any algrebraically closed field of characteristic 0, for simplicity all Lie algebras are taken to be complex.