Abstract
This chapter focuses on the characters of irreducible representations of semisimple Lie algebras. A recursion formula for the multiplicity of a weight of an irreducible representation is obtained. Let ρ be an irreducible representation of a semisimple Lie algebra g with highest weight ω0, ρ(G) be the Casimir operator of ρ and ρ(G) = rI. Formula of the character of an irreducible representation is also determined. Let g be a semisimple Lie algebra, h be a Cartan subalgebra, Σ be the system of roots and Π = {α1, …, αn} be a fundamental system of roots. By means of the character formula, H. Weyl also obtained the formula for the dimension of the corresponding representation space.
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