Character (mathematics): Difference between revisions
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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. |
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== |
==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. |
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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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>. |
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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==Examples== |
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*Consider the (''ax'' + ''b'')-group |
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:: <math> G := \left\{ \left. \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\ \right|\ a > 0,\ b \in \mathbf{R} \right\}.</math> |
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: Functions ''f''<sub>''u''</sub> : ''G'' → '''C''' such that <math>f_u \left(\begin{pmatrix} |
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a & b \\ |
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0 & 1 \end{pmatrix}\right)=a^u,</math> where ''u'' ranges over complex numbers '''C''' are multiplicative characters. |
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* Consider the multiplicative group of positive real numbers ('''R'''<sup>+</sup>,·). Then functions ''f''<sub>''u''</sub> : ('''R'''<sup>+</sup>,·) → '''C''' such that ''f''<sub>''u''</sub>(''a'') = ''a''<sup>''u''</sup>, where ''a'' is an element of ('''R'''<sup>+</sup>, ·) and ''u'' ranges over complex numbers '''C''', are multiplicative characters. |
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==Character of a representation== |
==Character of a representation== |
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Revision as of 00:59, 23 July 2012
If an internal link led you here, you may wish to change the link to point directly to the intended article.
In mathematics, a character is (most commonly) a special kind of function from a group to a 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.
Multiplicative character
A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field (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.
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 characters can be seen as a special case of this definition.
Multiplicative characters are linearly independent, i.e. if are different characters on a group G then from it follows that .
Character of a representation
The character of a representation φ of a group G on a finite-dimensional vector space V over a field F is the trace of the representation φ (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.
See also
- Dirichlet character
- Harish-Chandra character
- Hecke character
- Infinitesimal character
- Alternating character
References
- Artin, Emil (1966), Galois Theory, Notre Dame Mathematical Lectures, number 2, Arthur Norton Milgram (Reprinted Dover Publications, 1997), ISBN 978-0-486-62342-9
{{citation}}: Unknown parameter|note=ignored (help) - Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Springer-Verlag, ISBN 0-387-90190-6.