In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that arise in the related context of character theory. Whenever a group is represented by matrices, the function defined by the trace of the matrices is called a character; however, these traces do not in general form a group. Some important properties of these one-dimensional characters apply to characters in general:
Contents
The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the discrete Fourier transform. For locally compact abelian groups, the character group (with an assumption of continuity) is central to Fourier analysis.
Preliminaries
Let G be an abelian group. A function
If f is a character of a finite group G, then each function value f(g) is a root of unity (since
Each character f is a constant on conjugacy classes of G, that is, f(h g h−1) = f(g). For this reason, the character is sometimes called the class function.
A finite abelian group of order n has exactly n distinct characters. These are denoted by f1, ..., fn. The function f1 is the trivial representation; that is,
Definition
If G is an abelian group of order n, then the set of characters fk forms an abelian group under multiplication
Orthogonality of characters
Consider the
The sum of the entries in the jth row of A is given by
The sum of the entries in the kth column of A is given by
Let
This implies the desired orthogonality relationship for the characters: i.e.,
where