In mathematics, a character group is the group of representationss of an abelian group by complex-valued functionss. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characterss 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.

Preliminaries

is called a character if it is a group homomorphism.

is a character of a finite group (or more generally a torsion group is a root of unity.

A finite abelian group of order n has exactly n distinct characters. These are denoted by f1, ..., fn. The function f1. 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. This group is the character group of G is the principal character f1, and the inverse of a character fk is its reciprocal 1/fk is finite of order n is also of order n, the inverse of a character is equal to the complex conjugate.

Alternative definition

There is another definition of character grouppg 29. This is useful when studying complex tori is canonically isomorphic to the dual torus via the Appell–Humbert theorem. That is, can be expressed as. If we consider the lattice as a subgroup of the underlying real vector space, then a homomorphismcan be factored as a mapThis follows from elementary properties of homomorphisms. Note thatgiving us the desired factorization. As the group, we haveafter composing with the complex exponential, we find thatwhich is the expected result.

Examples

Finitely generated abelian groups

Since every finitely generated abelian group is isomorphic to is isomorphic to.

Orthogonality of characters

matrix A = A(G is the kth element of G.

The sum of the entries in the jth row of A is given by

1. , and

2. .

The sum of the entries in the kth column of A is given by

1. , and

2. .

denote the conjugate transpose of A. Then

1. .

This implies the desired orthogonality relationship for the characters: i.e.,

1. ,

is the Kronecker delta.

See also

• Pontryagin duality

References

• See chapter 6 of