Brauer character

Definition

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.