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.
to the group of non-zero complex numbers
is a character of a finite group (or more generally a torsion group)
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.
The function f1 is the trivial representation, which is given by
It is called the principal character of G; the others are called the non-principal characters.
If G is an abelian group, then the set of characters fk forms an abelian group under pointwise multiplication.
is the principal character f1, and the inverse of a character fk is its reciprocal 1/fk.
, the inverse of a character is equal to the complex conjugate.
There is another definition of character group[1]pg 29 which uses
This is useful when studying complex tori because the character group of the lattice in a complex torus
is canonically isomorphic to the dual torus via the Appell–Humbert theorem.
We can express explicit elements in the character group as follows: recall that elements in
If we consider the lattice as a subgroup of the underlying real vector space of
This follows from elementary properties of homomorphisms.
giving us the desired factorization.
after composing with the complex exponential, we find that
Since every finitely generated abelian group is isomorphic to
the character group can be easily computed in all finitely generated cases.
From universal properties, and the isomorphism between finite products and coproducts, we have the character groups of
, the second is computed by looking at the maps which send the generator
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