Given a finite group and a prime , we construct a field extension over satisfying the following properties. Let denote the ring of algebraic integers in , and be a prime ideal dividing the ideal .
- contains all roots of unity where is the index of the -Sylow subgroups in
- Every linear representation of over characteristic zero is realizable over
- Every representation of in characteristic is realizable over
We then define the Brauer character of a representation over as a map from the set of elements whose order is relatively prime to , to . The map is defined thus:
To any element of order relatively prime ot , diagonalize . Now map each eigenvalue (a root of unity in ) to the corresponding root of unity in , and take the sum of these. This number is the evaluation of the Brauer character of at .
The Brauer characters are important because of the following remarkable fact: the Brauer characters span the space of class functions on the -regular conjugacy classes (viz conjugacy classes of elements whose orders are relatively prime to ). In particular, we can get an explicit formula for expressing any character (restricted to the -regular conjugacy classes), as an integer linear combination of the Brauer characters.
The idea is to start off with a representation over and use it to obtain representations over by taking all matrix entries modulo the prime ideal. This idea needs to be smoothened out a bit to actually get a representation.
The coefficients are termed decomposition numbers. We can also look at the more general case of a number whose order may contain some power of , by first decomposing into its -regular and -singular part, and then evaluating for the -regular part.