Character group - Definition and Overview

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.

Preliminaries

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

If f is a character of a finite group G with identity e, then <math>f(e)=1<math> and each function value f(g) is a root of unity.

If 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, <math>f_1(g)=1 \;\; \forall g \in G<math>. It is called the principal character of G; the others are called the non-principal characters. The non-principal characters have the property that <math>f_i(g)\neq 1<math> for some <math>g \in G<math>.

Definition

If G is an abelian group, then the set of characters f_k forms an abelian group under multiplication <math>(f_j f_k)(g)= f_j(g) f_k(g)<math> for each element <math>g \in G<math>. This group is the character group of G and is sometimes denoted as <math>\hat {G}<math>. It is of order n. The identity element of <math>\hat {G}<math> is the principal character f₁. The inverse of f_k is the reciprocol 1/f_k. Note that since <math>|f_k(g)|=1 \;\; \forall g \in G<math>, that the inverse is equal to the complex conjugate.

Orthogonality of characters

Consider the <math>n \times n<math> matrix A=A(G) whose matrix elements are <math>A_{jk}=f_j(g_k)<math> where <math>g_k<math> is the kth element of G.

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

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

<math>\sum_{k=1}^n A_{1k} = n<math> for the case j=1.

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

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

<math>\sum_{j=1}^n A_{j1} = n<math> for the case g_k=e.

Let <math>A^\dagger<math> denote the conjugate transpose of A. Then

<math>AA^\dagger = A^\dagger A = nI<math>.

This implies the desired orthogonality relationship for the characters: that is

<math>\sum_{k=1}^n {f_k}^* (g_i) f_k (g_j) = n \delta_{ij}<math>

where <math>\delta_{ij}<math> is the kronecker delta and <math>f^*_k (g_i)<math> is the complex conjugate of <math>f_k (g_i)<math>.

Residue classes

Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k: <math>\hat{n}=\{m | m = n \mod k \}<math> That is, the residue class <math>\hat{n}<math> is the coset of n in the quotient group Z/kZ; it is an element of the cyclic group Z/kZ.

Given an integer k, one defines the set of reduced residue classes as the set <math>\{\hat{n}_1, \hat{n}_2, ... \hat{n}_{\phi(k)}\}<math> of residue classes that are coprime to k. This is the set of the generators of Z/kZ. The size of this set is obviously given by <math>\phi(k)<math>, Euler's totient phi. For example, for k=6, the set of reduced residue classes is <math>\{\hat{1}, \hat{5}\}<math> because 0, 2, 3, and 4 are not coprime to 6.

Theorem: The set of reduced residue classes modulo k forms an abelian group of order <math>\phi(k)<math> where group multiplication is given by <math>\hat{mn}=\hat{m}\hat{n}<math>. The identity is the residue class <math>\hat{1}<math> and the inverse of <math>\hat{m}<math> is the residue class <math>\hat{n}<math> where <math>mn=1 \mod k<math>.

Dirichlet characters

Let G be the group of reduced residue classes modulo k. Then, for each character f of G there exists an arithmetic function <math>\chi=\chi_f<math> defined as

<math>\chi(n)=f(\hat{n})<math> if <math>(n,k)=1<math> and

<math>\chi(n)=0<math> if <math>(n,k)>1<math>.

The function <math>\chi<math> is called a Dirichlet character modulo k. The principal character <math>\chi_1<math> has the properties

<math>\chi(n)=1<math> if <math>(n,k)=1<math> and

<math>\chi(n)=0<math> if <math>(n,k)>1<math>.

The properties of the functions <math>\chi<math> are developed further in the article on Dirichlet characters.

See also

• Primitive root modulo n

References

• Tom M. Apostol Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9 See chapter 6.