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{{For|the Reflection Group On  the Future of Europe|European Council}}
In [[group theory]] and [[geometry]], a '''reflection group''' is a [[discrete group]] which is generated by a set of [[reflection (mathematics)|reflections]] of a finite-dimensional [[Euclidean space]]. The symmetry group of a [[regular polytope]] or of a [[tessellation|tiling]] of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include [[Weyl group]]s and crystallographic [[Coxeter group]]s. While the [[orthogonal group]] is generated by reflections (by the [[Cartan–Dieudonné theorem]]), it is a continuous group (indeed, [[Lie group]]), not a discrete group, and is generally considered separately.
 
== Definition ==
 
Let ''E'' be a finite-dimensional [[Euclidean space]].  A '''finite reflection group''' is a subgroup of the [[general linear group]] of ''E'' which is generated by a set of orthogonal [[reflection (mathematics)|reflection]]s across hyperplanes passing through the origin. An '''affine reflection group''' is a discrete subgroup of the [[affine group]] of ''E'' that is generated by a set of ''affine reflections'' of ''E'' (without the requirement that the reflection hyperplanes pass through the origin).
 
The corresponding notions can be defined over other [[field (mathematics)|fields]], leading to '''[[complex reflection group]]s''' and analogues of reflection groups over a [[finite field]].
 
== Examples ==
=== Plane ===
 
In two dimensions, the finite reflection groups are the [[dihedral group]]s, which are generated by reflection in two lines that form an angle of <math>2\pi/n</math> and correspond to the [[Coxeter diagram]] <math>I_2(n).</math> Conversely, the cyclic [[point groups in two dimensions]] are ''not'' generated by reflections, and indeed contain no reflections – they are however subgroups of index 2 of a dihedral group.
 
Infinite reflection groups include the [[frieze group]]s <math>*\infty\infty</math> and <math>*22\infty</math> and the [[wallpaper group]]s <math>**</math>, <math>*2222</math>,<math>*333</math>,<math>*442</math> and <math>*632</math>. If the angle between two lines is an irrational multiple of pi, the group generated by reflections in these lines is infinite and non-discrete, hence, it is not a reflection group.
 
=== Space ===
 
Finite reflection groups are the [[point groups in three dimensions|point groups]] ''C<sub>nv</sub>'', ''D<sub>nh</sub>'', and the [[symmetry group]]s of the five [[Platonic solid]]s. Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups. The classification of finite reflection groups of '''R'''<sup>3</sup> is an instance of the [[ADE classification]].
 
== Kaleidoscopes ==
 
Reflection groups have deep relations with [[kaleidoscope]]s, as discussed in {{Harv|Goodman|2004}}.
 
== Relation with Coxeter groups ==
 
A reflection group ''W'' admits a [[group presentation|presentation]] of a special kind discovered and studied by [[H.S.M. Coxeter]]. The reflections in the faces of a fixed [[fundamental domain|fundamental]] "chamber" are generators ''r''<sub>''i''</sub> of ''W'' of order 2. All relations between them formally follow from the relations
 
: <math>(r_i r_j)^{c_{ij}}=1</math>
 
expressing the fact that the product of the reflections ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in two hyperplanes ''H''<sub>''i''</sub> and ''H''<sub>''j''</sub> meeting at an angle <math>\pi/c_{ij}</math> is a [[rotation]] by the angle <math>2\pi/c_{ij}</math> fixing the subspace ''H''<sub>''i''</sub>&nbsp;∩&nbsp;''H''<sub>''j''</sub> of codimension 2. Thus, viewed as an abstract group, every reflection group is a [[Coxeter group]].
 
== Finite fields ==
 
When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as <math>-1=1</math> so reflections are the identity).{{Citation needed|date=April 2010}} Geometrically, this amounts to including [[Shear mapping|shear]]s in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in {{Harv|Zalesskiĭ|Serežkin|1981}}.
 
== Generalizations ==
Discrete [[isometry group]]s of more general [[Riemannian manifold]]s generated by reflections have also been considered. The most important class arises from [[Riemannian symmetric space]]s of rank 1: the [[n-sphere]] ''S''<sup>''n''</sup>, corresponding to finite reflection groups, the Euclidean space '''R'''<sup>''n''</sup>, corresponding to
affine reflection groups, and the [[hyperbolic space]] ''H''<sup>''n''</sup>, where the corresponding groups are called '''hyperbolic reflection groups'''. In two dimensions, [[triangle group]]s include reflection groups of all three kinds.
 
== See also ==
 
* [[Hyperplane arrangement]]
* [[Chevalley–Shephard–Todd theorem]]
 
== References ==
 
Standard references include {{Harv|Humphreys|1992}} and {{Harv|Grove|Benson|1996}}.
 
{{Reflist}}
{{refbegin}}
* {{Citation
|authorlink=H.S.M. Coxeter
|first=H.S.M.
|last=Coxeter
|title=Discrete groups generated by reflections
|journal=Ann. of Math.
|volume=35
|year=1934
|pages=588–621
}}
* {{Citation
|authorlink=H.S.M. Coxeter
|first=H.S.M.
|last=Coxeter
|title=The complete enumeration of finite groups of the form <math>r_i^2=(r_ir_j)^{k_{ij}}=1</math>
|journal=J. London Math. Soc.
|volume=10
|year=1935
|pages=21–25
}}
*{{Citation
|first=Roe
|last=Goodman
|url=http://www.math.rutgers.edu/~goodman/pub/monthly.pdf
|title=The Mathematics of Mirrors and Kaleidoscopes
|journal=[[American Mathematical Monthly]]
|date=April 2004
}}
* {{Citation
|title=Reflection groups and Coxeter groups
|isbn=978-0-521-43613-7
|first=James E.
|last=Humphreys
|year=1992
|publisher=[[Cambridge University Press]]
}}
* {{Citation
|doi=10.1070/IM1981v017n03ABEH001369
|first1=A E
|last1=Zalesskiĭ
|first2=V N
|last2=Serežkin
|title=Finite Linear Groups Generated by Reflections
|journal=Math. USSR Izv.
|year=1981
|volume=17
|number=3
|pages=477–503
}}
*{{Citation
|url=http://www.cms.math.ca/Publications/Reviews/2003/rev4.pdf
|title=Reflection groups and invariant theory (review)
|first=Richard
|last=Kane
}}
*{{Citation
|arxiv=math/0405135
|title=Jacobians of reflection groups
|first1=Julia
|last1=Hartmann
|first2=Anne V.
|last2=Shepler
}}
*{{Citation
|arxiv=math.AG/0610938
|title=Reflection groups in algebraic geometry
|authorlink=Igor V. Dolgachev
|first=Igor V.
|last=Dolgachev
}}
{{refend}}
 
== External links ==
 
* {{eom|id=Reflection_group|title=Reflection group}}
 
[[Category:Reflection groups| ]]

Revision as of 06:31, 16 January 2014

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately.

Definition

Let E be a finite-dimensional Euclidean space. A finite reflection group is a subgroup of the general linear group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin. An affine reflection group is a discrete subgroup of the affine group of E that is generated by a set of affine reflections of E (without the requirement that the reflection hyperplanes pass through the origin).

The corresponding notions can be defined over other fields, leading to complex reflection groups and analogues of reflection groups over a finite field.

Examples

Plane

In two dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of and correspond to the Coxeter diagram Conversely, the cyclic point groups in two dimensions are not generated by reflections, and indeed contain no reflections – they are however subgroups of index 2 of a dihedral group.

Infinite reflection groups include the frieze groups and and the wallpaper groups , ,, and . If the angle between two lines is an irrational multiple of pi, the group generated by reflections in these lines is infinite and non-discrete, hence, it is not a reflection group.

Space

Finite reflection groups are the point groups Cnv, Dnh, and the symmetry groups of the five Platonic solids. Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups. The classification of finite reflection groups of R3 is an instance of the ADE classification.

Kaleidoscopes

Reflection groups have deep relations with kaleidoscopes, as discussed in Template:Harv.

Relation with Coxeter groups

A reflection group W admits a presentation of a special kind discovered and studied by H.S.M. Coxeter. The reflections in the faces of a fixed fundamental "chamber" are generators ri of W of order 2. All relations between them formally follow from the relations

expressing the fact that the product of the reflections ri and rj in two hyperplanes Hi and Hj meeting at an angle is a rotation by the angle fixing the subspace Hi ∩ Hj of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter group.

Finite fields

When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as so reflections are the identity).Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in Template:Harv.

Generalizations

Discrete isometry groups of more general Riemannian manifolds generated by reflections have also been considered. The most important class arises from Riemannian symmetric spaces of rank 1: the n-sphere Sn, corresponding to finite reflection groups, the Euclidean space Rn, corresponding to affine reflection groups, and the hyperbolic space Hn, where the corresponding groups are called hyperbolic reflection groups. In two dimensions, triangle groups include reflection groups of all three kinds.

See also

References

Standard references include Template:Harv and Template:Harv.

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    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.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    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

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    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
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    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.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    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

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    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
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    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.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    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

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    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

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