Difference between revisions of "Brauer character"
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Latest revision as of 23:12, 7 May 2008
Definition
Setup
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 .
Facts
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.
