We examine the relationship between the representations of a fixed group over different rings. Often we have have assumed that representations are defined over a field that is algebraically closed. What if the field is not algebraically closed? Such a question is significant because representations arise naturally over different fields that might not be algebraically closed, and it is important to know how they change on moving to an extension field, such as the algebraic closure. It is also important to know whether a representation may be defined over some smaller field. We introduce the notion of a splitting field, showing that such a field may always be chosen to be a finite extension of the prime field.
After proving Brauer's Theorem, that over a splitting field of characteristic p the number of nonisomorphic simple representations equals the number of conjugacy classes of elements of order prime to p, we turn to the question of reducing representations from characteristic 0 to characteristic p. The process involves first writing a representation in the valuation ring of a p-local field and then factoring out the maximal ideal of the valuation ring. This gives rise to the decomposition map between the Grothendieck groups of representations in characteristic 0 and characteristic p. We show that this map is well defined and then construct the so-called cde triangle. This provides a very effective way to compute the Cartan matrix in characteristic p from the decomposition map.
In the last part of this chapter, we describe in detail the properties of blocks of defect zero. These are representations in characteristic p that are both simple and projective. They always arise as the reduction modulo p of a simple representations in characteristic zero, and these are also known as blocks of defect 0. The blocks of defect zero have importance in character theory, accounting for many zeroes in character tables, and they are also the subject of some of the deepest investigations in representation theory.
Suppose that A is an algebra A over a commutative ring R and that U is an Amodule.