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 (with an assumption of continuity) is central to Fourier analysis.
Preliminaries.
Let "G" be an abelian group. A function formula_1 mapping the group to the non-zero complex numbers is called a character of "G" if it is a group homomorphism from formula_2 to formula_3—that is, if formula_4 for all formula_5.
If "f" is a character of a finite group "G", then each function value "f"("g") is a root of unity, since for each formula_6 there exists formula_7 such that formula_8, and hence formula_9.
Each character "f" is a constant on conjugacy classes of "G", that is, "f"("hgh"−1) = "f"("g"). 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 "f"1, ..., "f""n". The function "f"1 is the trivial representation, which is given by formula_10 for all formula_6. It is called the principal character of G; the others are called the non-principal characters.
Definition.
If "G" is an abelian group, then the set of characters "fk" forms an abelian group under pointwise multiplication. That is, the product of characters formula_12 and formula_13 is defined by formula_14 for all formula_6. This group is the character group of G and is sometimes denoted as formula_16. The identity element of formula_16 is the principal character "f"1, and the inverse of a character "fk" is its reciprocal 1/"fk". If formula_2 is finite of order "n", then formula_16 is also of order "n". In this case, since formula_20 for all formula_6, the inverse of a character is equal to the complex conjugate.
Orthogonality of characters.
Consider the formula_22 matrix "A" = "A"("G") whose matrix elements are formula_23 where formula_24 is the "k"th element of "G". 
The sum of the entries in the "j"th row of "A" is given by
The sum of the entries in the "k"th column of "A" is given by
Let formula_31 denote the conjugate transpose of "A". Then 
This implies the desired orthogonality relationship for the characters: i.e.,
where formula_34 is the Kronecker delta and formula_35 is the complex conjugate of formula_36.