Character group

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In mathematics, a character group is the group of representations of a group by complex-valued functions. The term character also arises in a different but related context, that of character theory. When a group is represented by matrices, the trace of the matrix is also called a character; however, these traces do not in general form a group. They do, however, share some important properties with the characters of the character group:

Characters are invariant on conjugacy classes.
The characters of an irreducible representation are orthogonal.

The primary importance of the character group 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.

1 Preliminaries
2 Definition
3 Orthogonality of characters
4 References

[edit] Preliminaries

Let G be an arbitrary group. A function $f:G\rightarrow \mathbb{C}\backslash\{0\}$ mapping the group to the non-zero complex numbers is called a character of G if it is a group homomorphism, that is, if $\forall g_1,g_2 \in G\;\; f(g_1 g_2)=f(g_1)f(g_2)$ . If the identity of the group is e, then $f (e) = 1$ .

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 f₁, ..., f_n. The function f₁ is the trivial representation; that is, $\forall g \in G\;\; f_1(g)=1$ . It is called the principal character of G; the others are called the non-principal characters. The non-principal characters have the property that $f_i(g)\neq 1$ for some $g \in G$ .

[edit] Definition

If G is an abelian group, 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 $g \in G$ . This group is the character group of G and is sometimes denoted as $\hat {G}$ . It is of order n. The identity element of $\hat {G}$ is the principal character f₁. The inverse of f_k is the reciprocal 1/f_k. Note that since $\forall g \in G\;\; |f_k(g)|=1$ , the inverse is equal to the complex conjugate.