Representations of Finite Groups

Abstract

The theory of group representations occurs in many contexts. First, it is developed for its own sake: determine all irreducible representations of a given group. See for instance Curtis-Reiner’s Methods of Representation Theory (Wiley-Interscience, 1981). It is also used in classifying finite simple groups. But already in this book we have seen applications of representations to Galois theory and the determination of the Galois group over the rationals. In addition, there is an analogous theory for topological groups. In this case, the closest analogy is with compact groups, and the reader will find a self-contained treatment of the compact case entirely similar to §5 of this chapter in my book SL ₂(R) (Springer Verlag), Chapter II, §2. Essentially, finite sums are replaced by integrals, otherwise the formalism is the same. The analysis comes only in two places. One of them is to show that every irreducible representation of a compact group is finite dimensional; the other is Schur’s lemma. The details of these extra considerations are carried out completely in the above-mentioned reference. I was careful to write up §5 with the analogy in mind.