ABSTRACT
In this first chapter, we present the general representation theory of finite groups. After an exposition of Maschke’s theorem of complete reducibility of representations (Section 1.1) and of Schur’s lemma of orthogonality of characters (Section 1.2), we construct the non-commutative Fourier transform (Section 1.3), which provides a decomposition of the complex group algebra ℂ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq37.tif"/> in blocks of endomorphism rings of the irreducible representations of G. It implies that any function f : G → ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq38.tif"/> can be expanded uniquely as a linear combination of the matrix coefficients of the irreducible representations of G (Proposition 1.15). This can be seen as a motivation for the study of representations of groups, and on the other hand, the Fourier isomorphism ℂ G → ⊕ λ ∈ G ˜ Eng ( V λ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq39.tif"/>
