Character Theory and the Orthogonality Relations

Abstract

This chapter gets to the heart of group representation theory: the character theory of Frobenius and Schur. The fundamental idea of character theory is to encode a representation $\phi : G\,\rightarrow \,{GL}_{n}(\mathbb{C})$ of G by a complex-valued function χ_ϕ:G→C. In other words, we replace a function to an n-dimensional space with a function to a 1-dimensional space. The characters turn out to form an orthonormal set with respect to the inner product on functions, a fact that we shall use to prove the uniqueness of the decomposition of a representation into irreducibles.