Daniel Martling: Introduction to Representation Theory of Finite Groups

Abstract.

This text serves as an introduction to representation theory of finite groups, beginning with a background in group theory and linear algebra before formally defining representations, some representations are shown to be irreducible, while others are composed as a direct sum of the irreducibles (Maschke’s Theorem 3.24), culminating in the proof of Schur’s Lemma 3.25, which ensure the uniqueness of these compositions. Finally, character theory is introduced, simplifying representation theory by focusing on the trace of matrices, providing systematic methods to identify and decompose any given representation. Examples to demonstrate the these theoretical concepts are some cyclic and symmetric groups of small order. In particular, \(C_3\), \(C_4\), \(C_5\), \(S_3\), and \(S_4\) are examined and completely decomposed and the results are presented in Tables 3, 4, 5, 13 and 14.