4 - Linear representations

Summary

Chapter 4 develops the elementary theory of linear representations. Linear representations are discussed from the point of view of modules over the group ring. Irreducibility and indecomposability are defined, and we find that the Jordan–Hölder Theorem holds for finite dimensional linear representations. Maschke's Theorem is established in section 12. Maschke's Theorem says that, if G is a finite group and F a field whose characteristic does not divide the order of G, then the indecomposable representations of G over F are irreducible.

Section 13 explores the connection between finite dimensional linear representations and matrices. There is also a discussion of the special linear group, the general linear group, and the corresponding projective groups. In particular we find that the special linear group is generated by its transvections and is almost always perfect.

Section 14 contains a discussion of the dual representation which will be needed in section 17.

Modules over the group ring

Section 12 studies linear representations over a field F using the group ring of G over F. This requires an elementary knowledge of modules over rings. One reference for this material is chapter 3 of Lang [La].

Throughout section 12, V will be a vector space over F. The group of automorphisms of V in the category of vector spaces and F-linear transformations is the general linear group GL(V). Assume π:G → GL(V) is a representation of G in this category.