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| 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 numbers]]). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified.
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| ==Multiplicative character==
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| {{main|multiplicative character}}
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| 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 numbers]]. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an [[abelian group]] under pointwise multiplication.
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| 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.
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| 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 + \ldots + a_n \chi_n = 0 </math> it follows that <math>a_1=a_2=\cdots=a_n=0 </math>.
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| ==Character of a representation==
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| {{main|Character theory}}
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| The '''character of a representation''' ''φ'' of a group ''G'' on a finite-dimensional [[vector space]] ''V'' over a field ''F'' is the [[trace (matrix)|trace]] of the [[group representation|representation]] ''φ'' {{Harv|Serre|1977}}. 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.
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| == See also ==
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| * [[Dirichlet character]]
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| * [[Harish-Chandra character]]
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| * [[Hecke character]]
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| * [[Infinitesimal character]]
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| * [[Alternating character]]
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| ==References==
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| * {{citation|title=Galois Theory|note= Lectures Delivered at the University of Notre Dame|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}}
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| * {{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}}.
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| ==External links==
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| * {{springer|title=Character of a group|id=p/c021560}}
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| * {{planetmath reference|id=1843|title=Character of a group representation}}
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| [[Category:Representation theory]]
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