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In [[mathematics]], a '''character group''' is the group of [[group representation|representation]]s of a [[group (mathematics)|group]] by  [[complex number|complex]]-valued [[function (mathematics)|functions]]. These functions can be thought of as one-dimensional [[matrix (mathematics)|matrix]] representations and so are special cases of the group [[character (mathematics)|characters]] that arise in the related context of [[character theory]]. Whenever a group is represented by [[matrix (mathematics)|matrices]], the function defined by the [[trace (matrix)|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 <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>\forall g_1,g_2 \in G\;\; f(g_1 g_2)=f(g_1)f(g_2)</math>.
 
If ''f'' is a character of a finite group ''G'', then each function value ''f(g)'' is a [[root of unity]] (since <math>\forall g \in G \;\; \exists k \in \mathbb{N}</math> such that <math>g^{k}=e</math>, <math>f(g)^{k}=f(g^{k})=f(e)=1</math>).
 
Each character ''f'' is a constant on [[conjugacy class]]es of ''G'', that is, ''f''(''h'' ''g'' ''h''<sup>−1</sup>) = ''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 ''f''<sub>1</sub>, ..., ''f''<sub>n</sub>.  The function ''f''<sub>1</sub> is the trivial representation; that is, <math>\forall g \in G\;\; f_1(g)=1</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 of order ''n'', then the set of characters ''f<sub>k</sub>'' forms an abelian group under multiplication <math>(f_j f_k)(g)= f_j(g) f_k(g)</math> <!-- this notation assumes the character group is automatically finite for some reason. ---> 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 ''f''<sub>1</sub>.  The inverse of ''f''<sub>k</sub> is the reciprocal 1/''f''<sub>k</sub>.  Note that since <math>\forall g \in G\;\; |f_k(g)|=1</math>, 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 ''k''th element of ''G''.
 
The sum of the entries in the ''j''th 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>.
 
The sum of the entries in the ''k''th 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} = \sum_{j=1}^n f_j(e) = n</math>.
 
Let <math>A^\ast</math> denote the [[conjugate transpose]] of ''A''.  Then
:<math>AA^\ast = A^\ast A = nI</math>.
This implies the desired orthogonality relationship for the characters: i.e.,
 
:<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>.
 
== See also ==
 
* [[Pontryagin duality]]
 
==References==
* See chapter 6 of {{Apostol IANT}}
 
[[Category:Number theory]]
[[Category:Group theory]]
[[Category:Representation theory of groups]]

Revision as of 14:26, 25 January 2014

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{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 g1,g2Gf(g1g2)=f(g1)f(g2).

If f is a character of a finite group G, then each function value f(g) is a root of unity (since gGk such that gk=e, f(g)k=f(gk)=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, ..., fn. The function f1 is the trivial representation; that is, gGf1(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 fi(g)1 for some gG.

Definition

If G is an abelian group of order n, then the set of characters fk forms an abelian group under multiplication (fjfk)(g)=fj(g)fk(g) for each element gG. This group is the character group of G and is sometimes denoted as G^. It is of order n. The identity element of G^ is the principal character f1. The inverse of fk is the reciprocal 1/fk. Note that since gG|fk(g)|=1, the inverse is equal to the complex conjugate.

Orthogonality of characters

Consider the n×n matrix A=A(G) whose matrix elements are Ajk=fj(gk) where gk is the kth element of G.

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

k=1nAjk=k=1nfj(gk)=0 if j1, and
k=1nA1k=n.

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

j=1nAjk=j=1nfj(gk)=0 if k1, and
j=1nAj1=j=1nfj(e)=n.

Let A denote the conjugate transpose of A. Then

AA=AA=nI.

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

k=1nfk*(gi)fk(gj)=nδij ,

where δij is the Kronecker delta and fk*(gi) is the complex conjugate of fk(gi).

See also

References