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.