Character (mathematics)
293418
210793264
2008-05-07T12:49:27Z
Robert K S
168790
{{other uses|character}}
There are several meanings of the word '''''character''''' in mathematics, although all are related to the idea of using [[field (mathematics)|fields]] (most of the time the [[complex numbers]]) to study a more abstract algebraic structure.
==Number-theoretic characters==
If ''G'' is a [[group (mathematics)|group]], a character is a [[group homomorphism]] into the [[group (mathematics)|multiplicative group]] of a field (as defined in Emil Artin's book on Galois Theory), usually the field of [[complex numbers]]. If ''A'' is an [[abelian group]], then the set Ch(''A'') of these morphisms forms a group under the operation
:χ<sub>a</sub>χ<sub>b</sub>=χ<sub>ab</sub>.
This group is referred to as the [[character group]]. Sometimes only ''unitary'' characters are considered (so that the image is in the [[unit circle]]); other such homomorphisms are then called ''quasi-characters''. [[Dirichlet character]]s can be seen a special case of this definition.
==Representation characters==
If ''f'' is a finite-dimensional [[group representation|representation]] of a [[group (mathematics)|group]] ''G'', then the character of the representation is the function from ''G'' to the complex numbers given by the [[trace (matrix)|trace]] of ''f''. In general, the trace is neither a group homomorphism, nor does the set of traces form a group. The study of representations by means of their characters is called [[character theory]].
==Algebraic characters==
If ''A'' is an [[abelian algebra]] over the complex numbers, a character of ''A'' is an [[Algebra over a field|algebra homomorphism]] into the complex numbers. If in addition, ''A'' is a [[star-algebra|*-algebra]], then a character is a [[star-algebra|*-homomorphism]] into the complex numbers.
==External links==
* {{planetmath reference|id=1843|title=Character of a group representation}}
[[Category:Representation theory]]
[[Category:Mathematical disambiguation]]
[[de:Charakter (Mathematik)]]
[[fr:Caractère (mathématiques)]]
[[ru:Характер (теория групп)]]