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:

• 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

Let G be an abelian 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 from $G$ to $\mathbb C^\times$—that is, if $\forall g_1,g_2 \in G\;\; f(g_1 g_2)=f(g_1)f(g_2)$.

If f is a character of a finite group G, then each function value f(g) is a root of unity (since $\forall g \in G \;\; \exists k \in \mathbb{N}$ such that $g^{k}=e$, $f(g)^{k}=f(g^{k})=f(e)=1$).

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 $g \in G$.

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 $g \in G$. 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.

Orthogonality of characters

Consider the $n \times n$ 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

$\sum_{k=1}^n A_{jk} = \sum_{k=1}^n f_j(g_k) = 0$ if $j \neq 1$, and

$\sum_{k=1}^n A_{1k} = n$.

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

$\sum_{j=1}^n A_{jk} = \sum_{j=1}^n f_j(g_k) = 0$ if $k \neq 1$, and

$\sum_{j=1}^n A_{j1} = \sum_{j=1}^n f_j(e) = n$.

Let $A^\ast$ denote the conjugate transpose of A. Then

$AA^\ast = A^\ast A = nI$.

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

$\sum_{k=1}^n {f_k}^* (g_i) f_k (g_j) = n \delta_{ij}$ ,

where $\delta_{ij}$ is the Kronecker delta and $f^*_k (g_i)$ is the complex conjugate of $f_k (g_i)$.

See also

• Pontryagin duality

References

• See chapter 6 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001