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:
- Characters are invariant on conjugacy classes.
- The characters of irreducible representations are orthogonal.
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.
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 .
If f is a character of a finite group G, then each function value f(g) is a root of unity (since such that , ).
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, . It is called the principal character of G; the others are called the non-principal characters. The non-principal characters have the property that for some .
If G is an abelian group of order n, then the set of characters fk forms an abelian group under multiplication for each element . This group is the character group of G and is sometimes denoted as . It is of order n. The identity element of is the principal character f1. The inverse of fk is the reciprocal 1/fk. Note that since , the inverse is equal to the complex conjugate.
Orthogonality of characters
Consider the matrix A=A(G) 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
- if , and
The sum of the entries in the kth column of A is given by
- if , and
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 .