Chemistry - Character group

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.

Contents

1 Preliminaries

2 Definition

3 Orthogonality of characters

4 Residue classes

5 Dirichlet characters

6 See also

7 References

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 $f(g_1 g_2)=f(g_1)f(g_2) \;\;\forall g_1,g_2 \in G$ .

If f is a character of a finite group G with identity e, then $f (e) = 1$ 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, $f_1(g)=1 \;\; \forall g \in G$ . 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, 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 $|f_k(g)|=1 \;\; \forall g \in G$ , that the inverse is equal to the complex conjugate.

Orthogonality of characters

Consider the $n \times n$ matrix A=A(G) whose matrix elements are $A j k = f j (g k)$ 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

$\sum_{k=1}^n A_{1k} = n$ for the case j=1.

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

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

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

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

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

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

where $δ i j$ is the kronecker delta and $f^*_k (g_i)$ is the complex conjugate of $f k (g i)$ .

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: $\hat{n}=\{m | m = n \mod k \}.$ That is, the residue class $\hat{n}$ 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 $\{\hat{n}_1, \hat{n}_2, ... \hat{n}_{\phi(k)}\}$ 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 $φ(k)$ , Euler's totient phi. For example, for k=6, the set of reduced residue classes is $\{\hat{1}, \hat{5}\}$ 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 $φ(k)$ where group multiplication is given by $\hat{mn}=\hat{m}\hat{n}$ . The identity is the residue class $\hat{1}$ and the inverse of $\hat{m}$ is the residue class $\hat{n}$ where $mn=1 \mod k$ .