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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. | |||
==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 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 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 + \ldots + 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 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. | |||
== See also == | |||
* [[Dirichlet character]] | |||
* [[Harish-Chandra character]] | |||
* [[Hecke character]] | |||
* [[Infinitesimal character]] | |||
* [[Alternating character]] | |||
==References== | |||
* {{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}} | |||
* {{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}}. | |||
==External links== | |||
* {{springer|title=Character of a group|id=p/c021560}} | |||
* {{planetmath reference|id=1843|title=Character of a group representation}} | |||
[[Category:Representation theory]] | |||
Revision as of 15:25, 15 November 2013
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
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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 multiplicative group of a field Template:Harv, 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
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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 Template:Harv. 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
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
External links
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