After the success of ordinary character theory in computing with representations over fields of characteristic zero we wish for a corresponding theory of characters of representations in positive characteristic. Brauer characters provide such a theory and we describe it in this chapter, starting with the definition and continuing with their main properties, which are similar in many respects to those of ordinary characters. We arrange the Brauer characters in tables that satisfy orthogonality relations, although of a more complicated kind than what we saw with ordinary characters. We will find that the information carried by Brauer characters is exactly that of the composition factors of a representation, and not more. They are very effective if that is the information we require, but they do not tell us anything more about the range of complicated possibilities that we find with representations in positive characteristic. Using Brauer characters is, however, usually the bestway to compute a decompositionmatrix, and hence a Cartan matrix in view of Corollary 9.5.6.We obtain in Corollary 10.2.4 the deductions that the Cartanmatrix is invertible and the decompositionmatrix has maximum rank.

The Definition of Brauer Characters

Whenever we work with Brauer characters for a finite group G in characteristic p we will implicitly assume that we have a p-modular system (F, R, k) where both F and k are splitting fields for G and all of its subgroups. This has the implication that F and k both contain a primitive ath root of unity, where a is the l.c.m. of the orders of the p-regular elements of G (the elements of order prime to p). If we wish to define the Brauer character of a kG-moduleU where k or F do not contain a primitive ath root of unity, we first extend the scalars so that the p-modular system does have this property.

We now examine the relationship between the roots of unity in F and in k. Let us put

Lemma 10.1.1.With this notation,

1. (1) μF ⊆ R, and

2. (2) the quotient homomorphism R → R/(π) = k gives an isomorphism μF → μk.