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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:
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* 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==
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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]]

Latest revision as of 21:18, 20 October 2014

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