 Character group

In mathematics, a character group is the group of representations of a group by complexvalued functions. These functions can be thought of as onedimensional matrix representations and so are special cases of the group characters which arises 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 onedimensional 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.
Contents
Preliminaries
Let G be an abelian group. A function mapping the group to the nonzero complex numbers is called a character of G if it is a group homomorphism—that is, if and f(e) = 1 where e is the identity of the group.
If f is a character of a finite group G, then each function value f(g) is a root of unity (since all elements of a finite group have finite order).
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 f_{1}, ..., f_{n}. The function f_{1} is the trivial representation; that is, . It is called the principal character of G; the others are called the nonprincipal characters. The nonprincipal characters have the property that for some .
Definition
If G is an abelian group of order n, then the set of characters f_{k} forms an abelian group under multiplication (f_{j}f_{k})(g) = f_{j}(g)f_{k}(g) 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 f_{1}. The inverse of f_{k} is the reciprocal 1/f_{k}. Note that since , the inverse is equal to the complex conjugate.
Orthogonality of characters
Consider the matrix A=A(G) whose matrix elements are A_{jk} = f_{j}(g_{k}) where g_{k} 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 δ_{ij} is the Kronecker delta and is the complex conjugate of f_{k}(g_{i}).
See also
References
 See chapter 6 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New YorkHeidelberg: SpringerVerlag, ISBN 9780387901633, MR0434929
Categories: Number theory
 Group theory
 Representation theory of groups
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