Brauer character

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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.