Editing Character (mathematics)

In [[mathematics]], a '''character''' is (most commonly) a special kind of [[function (mathematics)|function]] from a [[group (mathematics)|group]] to a [[field (mathematics)|field]] (such as the [[complex number]]s). There are at least two distinct, but overlapping meanings.<ref>{{Cite web|url=https://ncatlab.org/nlab/show/character|title=character in nLab|website=ncatlab.org|access-date=2017-10-31}}</ref> Other uses of the word "character" are almost always qualified.

== Multiplicative character ==
{{main|multiplicative character}}

A '''multiplicative character''' (or '''linear character''', or simply '''character''') on a group ''G'' is a [[group homomorphism]] from ''G'' to the [[unit group|multiplicative group]] of a field {{Harv|Artin|1966}}, usually the field of [[complex number]]s.  If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an [[abelian group]] under pointwise multiplication.

This group is referred to as the [[character group]] of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the [[unit circle]]); other such homomorphisms are then called ''quasi-characters''. [[Dirichlet character]]s can be seen as a special case of this definition.

Multiplicative characters are [[linear independence|linearly independent]], i.e. if <math>\chi_1,\chi_2, \ldots , \chi_n </math> are different characters on a group ''G'' then from <math>a_1\chi_1+a_2\chi_2 + \dots + a_n \chi_n = 0 </math> it follows that <math>a_1=a_2=\cdots=a_n=0 </math>.

== Character of a representation ==
{{main|Character theory}}
The '''character''' <math>\chi : G \to F</math>  of a [[group representation|representation]] <math>\phi \colon G\to\mathrm{GL}(V)</math> of a group ''G'' on a [[dimension (vector space)|finite-dimensional]] [[vector space]] ''V'' over a field ''F'' is the [[trace (matrix)|trace]] of the representation <math>\phi</math> {{Harv|Serre|1977}}, i.e. 

:<math>\chi_\phi(g) = \operatorname{Tr}(\phi(g))</math> for <math>g \in G</math>

In general, the trace is not a group homomorphism, nor does the set of traces form a group.  The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher-dimensional characters. The study of representations using characters is called "[[character theory]]" and one-dimensional characters are also called "linear characters" within this context.

=== Alternative definition ===
If restricted to [[finite group|finite]] abelian group with <math>1 \times 1</math> representation in <math>\mathbb{C}</math> (i.e. <math>\mathrm{GL}(V) = \mathrm{GL}(1, \mathbb{C})</math>), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a [[direct sum]] of <math>1 \times 1</math> representations. For non-abelian groups, the original definition would be more general than this one):

A character <math>\chi</math> of group <math>(G, \cdot)</math> is a group homomorphism <math>\chi: G \rightarrow \mathbb{C}^*</math> i.e. <math> \chi (x \cdot y)=\chi (x) \chi (y)</math> for all <math> x, y \in G.</math>

If <math>G</math> is a finite abelian group, the characters play the role of [[Harmonic analysis|harmonics]]. For infinite abelian groups, the above would be replaced by <math>\chi: G \to \mathbb{T}</math> where <math>\mathbb{T}</math> is the [[circle group]].

== See also ==
* [[Character group]]
* [[Dirichlet character]]
* [[Harish-Chandra character]]
* [[Hecke character]]
* [[Infinitesimal character]]
* [[Alternating character]]
* [[Characterization (mathematics)]]
* [[Pontryagin duality]]
* {{slink|Base (topology)#Weight and character}}

== References ==
{{reflist}}
* {{citation|title=Galois Theory|series=Notre Dame Mathematical Lectures, number 2|authorlink=Emil Artin|first=Emil|last= Artin|year=1966|publisher = Arthur Norton Milgram (Reprinted Dover Publications, 1997)|isbn=978-0-486-62342-9}}  Lectures Delivered at the University of Notre Dame
* {{citation | authorlink=J.-P. Serre | first=Jean-Pierre | last=Serre | title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | isbn=0-387-90190-6 | location=New York-Heidelberg | series=[[Graduate Texts in Mathematics]] | volume=42 | others=Translated from the second French edition by Leonard L. Scott | mr=0450380 | doi=10.1007/978-1-4684-9458-7 | url-access=registration | url=https://archive.org/details/linearrepresenta1977serr }}

== External links ==
* {{springer|title=Character of a group|id=p/c021560}}

{{set index article|mathematics}}
[[Category:Representation theory]]