# Character (mathematics)

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Character_(mathematics)
> Markdown URL: https://mediated.wiki/source/Character_(mathematics).md
> Source: https://en.wikipedia.org/wiki/Character_(mathematics)
> Source revision: 1356570593
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Mathematical function}}
In [mathematics](/source/mathematics), a '''character''' is (most commonly) a special kind of [function](/source/function_(mathematics)) from a [group](/source/group_(mathematics)) to a [field](/source/field_(mathematics)) (such as the [complex number](/source/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](/source/group_homomorphism) from ''G'' to the [multiplicative group](/source/unit_group) of a field {{Harv|Artin|1966}}, usually the field of [complex number](/source/complex_number)s.  If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an [abelian group](/source/abelian_group) under pointwise multiplication.

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

Multiplicative characters are [linearly independent](/source/linear_independence), i.e. if <math>\chi_1,\chi_2, \ldots , \chi_n </math> are different characters on a group ''G'' to the same field 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 [representation](/source/group_representation) <math>\phi \colon G\to\mathrm{GL}(V)</math> of a group ''G'' on a [finite-dimensional](/source/dimension_(vector_space)) [vector space](/source/vector_space) ''V'' over a field ''F'' is the [trace](/source/trace_(matrix)) 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](/source/character_theory)" and one-dimensional characters are also called "linear characters" within this context.

=== Alternative definition ===
If restricted to [finite](/source/finite_group) 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](/source/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 [harmonics](/source/Harmonic_analysis). 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](/source/circle_group).

== See also ==
* [Character group](/source/Character_group)
* [Dirichlet character](/source/Dirichlet_character)
* [Harish-Chandra character](/source/Harish-Chandra_character)
* [Hecke character](/source/Hecke_character)
* [Infinitesimal character](/source/Infinitesimal_character)
* [Alternating character](/source/Alternating_character)
* [Characterization (mathematics)](/source/Characterization_(mathematics))
* [Pontryagin duality](/source/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](/source/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](/source/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}}

Category:Representation theory

---
Adapted from the Wikipedia article [Character (mathematics)](https://en.wikipedia.org/wiki/Character_(mathematics)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Character_(mathematics)?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.