Brauer character

Setup
Given a finite group $$G$$ and a prime $$p$$, we construct a field extension $$F$$ over $$\mathbb{Q}$$ satisfying the following properties. Let $$I$$ denote the ring of algebraic integers in $$F$$, and $$P$$ be a prime ideal dividing the ideal $$pI$$.


 * $$F$$ contains all $$g'^{th}$$ roots of unity where $$g'$$ is the index of the $$p$$-Sylow subgroups in $$G$$
 * Every linear representation of $$G$$ over characteristic zero is realizable over $$F$$
 * Every representation of $$G$$ in characteristic $$p$$ is realizable over $$K$$

We then define the Brauer character of a representation $$\sigma$$ over $$K$$ as a map from the set of elements whose order is relatively prime to $$p$$, to $$F$$. The map is defined thus:

To any element $$g$$ of order relatively prime ot $$p$$, diagonalize $$\sigma(g)$$. Now map each eigenvalue (a root of unity in $$K$$) to the corresponding root of unity in $$F$$, and take the sum of these. This number is the evaluation of the Brauer character of $$\sigma$$ at $$g$$.

Facts
The Brauer characters are important because of the following remarkable fact: the Brauer characters span the space of class functions on the $$p$$-regular conjugacy classes (viz conjugacy classes of elements whose orders are relatively prime to $$p$$). In particular, we can get an explicit formula for expressing any character (restricted to the $$p$$-regular conjugacy classes), as an integer linear combination of the Brauer characters.

The idea is to start off with a representation over $$F$$ and use it to obtain representations over $$K$$ 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 $$p$$, by first decomposing into its $$p$$-regular and $$p$$-singular part, and then evaluating for the $$p$$-regular part.