Representations of finite groups and semisimple algebras
Pierre‐Loïc Méliot · 2017
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"/>