Editing Character group

In [[mathematics]], a '''character group''' is the [[group (mathematics)|group]] of [[group representation|representation]]s of an [[abelian group]] by [[complex number|complex]]-valued [[function (mathematics)|function]]s. These functions can be thought of as one-dimensional [[matrix (mathematics)|matrix]] representations and so are special cases of the group [[character (mathematics)|character]]s that arise in the related context of [[character theory]]. Whenever a group is represented by matrices, the function defined by the [[trace (linear algebra)|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 class]]es.
* The characters of irreducible representations are orthogonal.
The primary importance of the character group for [[finite group|finite]] abelian groups is in [[number theory]], where it is used to construct [[Dirichlet character]]s.  The character group of the [[cyclic group]] also appears in the theory of the [[discrete Fourier transform]]. For [[locally compact abelian group]]s, the character group (with an assumption of continuity) is central to [[Fourier analysis]].

==Preliminaries==
{{Main|Character (mathematics)}}
Let <math>G</math> be an abelian group. A function <math>f: G \to \mathbb{C}^\times</math> mapping <math>G</math> to the group of non-zero complex numbers <math>\mathbb{C}^\times = \mathbb{C}\setminus\{0\}</math> is called a '''character''' of <math>G</math> if it is a [[group homomorphism]]—that is, if <math>f(g_1 g_2) = f(g_1)f(g_2)</math> for all <math>g_1, g_2 \in G</math>.

If <math>f</math> is a character of a finite group (or more generally a [[torsion group]]) <math>G</math>, then each function value <math>f(g)</math> is a [[root of unity]], since for each <math>g \in G</math> there exists <math>k \in \mathbb{N}</math> such that <math>g^k = e</math>, and hence <math>f(g)^k = f(g^k) = f(e) = 1</math>.

Each character ''f'' is a constant on conjugacy classes of ''G'', that is, ''f''(''hgh''<sup>−1</sup>) = ''f''(''g'').  For this reason, a character is sometimes called a '''class function'''.

A finite abelian group of [[order (group theory)|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, which is given by <math>f_1(g) = 1</math> for all <math>g \in G</math>.  It is called the '''principal character of ''G'''''; the others are called the '''non-principal characters'''.

== Definition ==
If ''G'' is an abelian group, then the set of characters ''f<sub>k</sub>'' forms an abelian group under pointwise multiplication. That is, the product of characters <math>f_j</math> and <math>f_k</math> is defined by <math>(f_j f_k)(g) = f_j(g) f_k(g)</math> for all <math>g \in G</math>.  This group is the '''character group of ''G''''' and is sometimes denoted as <math>\hat{G}</math>.  The identity element of <math>\hat{G}</math> is the principal character ''f''<sub>1</sub>, and the inverse of a character ''f<sub>k</sub>'' is its reciprocal 1/''f<sub>k</sub>''.  If <math>G</math> is finite of order ''n'', then <math>\hat{G}</math> is also of order ''n''.  In this case, since <math>|f_k(g)| = 1</math> for all <math>g \in G</math>, the inverse of a character is equal to the [[complex conjugate]].

=== Alternative definition ===
There is another definition of character group<ref>{{Cite book|last=Birkenhake|first=Christina |title=Complex Abelian varieties|date=2004|publisher=Springer|author2=H. Lange |isbn=3-540-20488-1|edition=2nd, augmented |location=Berlin|oclc=54475368}}</ref><sup>pg 29</sup> which uses <math>U(1) = \{z \in \mathbb{C}^*: |z|=1 \}</math> as the target instead of just <math>\mathbb{C}^*</math>. This is useful when studying [[complex torus|complex tori]] because the character group of the lattice in a complex torus <math>V/\Lambda</math> is [[canonical isomorphism|canonically isomorphic]] to the dual torus via the [[Appell–Humbert theorem]]. That is,<blockquote><math>\text{Hom}(\Lambda, U(1)) \cong V^\vee\!/\Lambda^\vee = X^\vee</math></blockquote>We can express explicit elements in the character group as follows: recall that elements in <math>U(1)</math> can be expressed as<blockquote><math>e^{2\pi i x}</math></blockquote>for <math>x \in \mathbb{R}</math>. If we consider the lattice as a [[subgroup]] of the underlying [[real number|real]] [[vector space]] of <math>V</math>, then a homomorphism<blockquote><math>\phi: \Lambda \to U(1)</math></blockquote>can be factored as a map<blockquote><math>\phi : \Lambda \to \mathbb{R} \xrightarrow{\exp({2\pi i \cdot })} U(1)</math></blockquote>This follows from elementary properties of homomorphisms. Note that<blockquote><math>\begin{align}
\phi(x+y) &= \exp({2\pi i }f(x+y)) \\
&= \phi(x) + \phi(y) \\
&= \exp(2\pi i f(x))\exp(2\pi i f(y))
\end{align}</math></blockquote>giving us the desired factorization. As the group<blockquote><math>\text{Hom}(\Lambda,\mathbb{R}) \cong \text{Hom}(\mathbb{Z}^{2n},\mathbb{R})</math></blockquote>we have the isomorphism of the character group, as a group, with the group of homomorphisms of <math>\mathbb{Z}^{2n}</math> to <math>\mathbb{R}</math>. Since <math>\text{Hom}(\mathbb{Z},G)\cong G</math> for any abelian group <math>G</math>, we have<blockquote><math>\text{Hom}(\mathbb{Z}^{2n}, \mathbb{R}) \cong \mathbb{R}^{2n}</math></blockquote>after composing with the complex exponential, we find that<blockquote><math>\text{Hom}(\mathbb{Z}^{2n}, U(1)) \cong \mathbb{R}^{2n}/\mathbb{Z}^{2n}</math></blockquote>which is the expected result.

== Examples ==

=== Finitely generated abelian groups ===
Since every [[finitely generated abelian group]] is isomorphic to<blockquote><math>G \cong \mathbb{Z}^n \oplus \bigoplus_{i=1}^m \mathbb{Z}/a_i</math></blockquote>the character group can be easily computed in all finitely generated cases. From universal properties, and the isomorphism between finite products and coproducts, we have the character groups of <math>G</math> is isomorphic to<blockquote><math>\text{Hom}(\mathbb{Z},\mathbb{C}^*)^{\oplus n}\oplus\bigoplus_{i=1}^k\text{Hom}(\mathbb{Z}/n_i,\mathbb{C}^*)</math></blockquote>for the first case, this is isomorphic to <math>(\mathbb{C}^*)^{\oplus n}</math>, the second is computed by looking at the maps which send the generator <math>1 \in \mathbb{Z}/n_i</math> to the various powers of the <math>n_i</math>-th roots of unity <math>\zeta_{n_i} = \exp(2\pi i/n_i)</math>.

== 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==
<references />
* See chapter 6 of {{Apostol IANT}}

[[Category:Number theory]]
[[Category:Group theory]]
[[Category:Representation theory of groups]]