Irreducible Characters and Weights

Abstract

The central result of this chapter is the Weyl character formula. It establishes a bijection between the irreducible characters of a compact connected Lie group and the integral forms in a distinguished Weyl chamber. The character formula is stated and proved in the first section. In the second section we will introduce partial orderings on LT* and analyze the structure of the character ring. The third section gives us some efficient formulas for computing the multiplicity of a weight in an irreducible representation. Finally, come the explicit calculations of representations and representation rings of classical groups.