Character group


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:
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 is central to Fourier analysis.

Preliminaries

Let G be an abelian group. A function mapping the group to the non-zero complex numbers is called a character of G if it is a group homomorphism from to —that is, if for all.
If f is a character of a finite group G, then each function value f is a root of unity, since for each there exists such that, and hence.
Each character f is a constant on conjugacy classes of G, that is, f = f. For this reason, a character is sometimes called a 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, which is given by for all. It is called the principal character of G; the others are called the non-principal characters.

Definition

If G is a abelian group, then the set of characters fk forms an abelian group under pointwise multiplication. That is, the product of characters and is defined by for all. This group is the character group of G and is sometimes denoted as. The identity element of is the principal character f1, and the inverse of a character fk is its reciprocal 1/fk. If is finite of order n, then is also of order n. In this case, since for all, the inverse of a character is equal to the complex conjugate.

Orthogonality of characters

Consider the matrix A = A whose matrix elements are where is the kth element of G.
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 denote the conjugate transpose of A. Then
This implies the desired orthogonality relationship for the characters: i.e.,
where is the Kronecker delta and is the complex conjugate of.