Lie sphere geometry - Revision history

en>Decora: /* See Also */

2014-07-30T22:40:18Z

en>Wavelength: inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

2014-02-20T03:33:06Z

inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

← Older revision		Revision as of 05:33, 20 February 2014
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	In [[mathematics]], especially in the area of [[abstract algebra\|algebra]] known as [[representation theory]], the '''representation ring''' (or '''Green ring''' after [[J. A. Green]]) of a [[group (mathematics)\|group]] is a [[Ring (mathematics)\|ring]] formed from all the (isomorphism classes of the) finite-dimensional linear [[group representation\|representations]] of the group. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of [[algebraically closed field]]s of [[characteristic of a ring\|characteristic]] ''p'' where the [[Sylow subgroup\|Sylow ''p''-subgroups]] are [[cyclic group\|cyclic]] is also theoretically approachable.		I am Alva and was born on 3 February 1970. My hobbies are Cheerleading and Rock stacking.<br><br>my site ... [http://tinyurl.com/nrfwlnd nike air max]

	~~==Formal definition==~~
	~~Given a group ''G'' and a field ''F'', the elements of its '''representation ring''' ''R''<sub>''F''</sub>(''G'')~~ are ~~the formal differences of isomorphism classes of finite dimensional linear ''F''-representations of ''G''. For the ring structure, addition is given by the direct sum of representations,~~ and ~~multiplication by their [[tensor product]] over ''F''. When ''F'' is omitted from the notation, as in ''R''(''G''), then ''F'' is implicitly taken to be the field of complex numbers.~~

	~~==Examples==~~
	*For the complex representations of the [[cyclic group]] of order ''n'', the representation ring ''R''<sub>'''''C'''''</sub>(''C''<sub>''n''</sub>) is isomorphic to '''Z'''[''X'']/(''X''<sup>''n''</sup> − 1), where ''X'' corresponds to the complex representation sending a generator of the group to a primitive ''n''th root of unity.
	*More generally, the complex representation ring of a finite [[abelian group]] may be identified with the [[group ring]] of the [[character group]].
	*For the rational representations of the cyclic group of order 3, the representation ring ''R''<sub>'''''Q'''''</sub>(C<sub>3</sub>) is isomorphic to '''''Z'''''[''X'']/(''X''<sup>2</sup> − ''X'' − 2), where ''X'' corresponds to the irreducible rational representation of dimension 2.
	*For the modular representations of the cyclic group of order 3 over a field ''F'' of characteristic 3, the representation ring ''R''<sub>''F''</sub>(''C''<sub>3</sub>) is isomorphic to '''''Z'''''[''X'',''Y'']/(''X''<sup>2</sup> − ''Y'' − 1, ''XY'' − 2''Y'',''Y''<sup>2</sup> − 3''Y'').

	*The ring ''R''(S<sup>1</sup>) for the circle group is isomorphic to '''''Z'''''[''X'', ''X''<~~sup~~>~~ −1~~<~~/sup~~>]. ~~The ring of real representations is the subring of ''R''(''G'') of elements fixed by the involution on ''R''(''G'') given by ''X'' → ''X''<sup> −1</sup>~~.

	*The ring ''R''<sub>'''''C'''''</sub>(''S''<sub>3</sub>) for the [[symmetric group]] on three points is isomorphic to '''Z'''[''X'',''Y'']/(''XY'' − ''Y'',''X''<sup>2</sup> − 1,''Y''<sup>2</sup> − ''X'' − ''Y'' − 1), where ''X'' is the 1-dimensional alternating representation and ''Y'' the 2-dimensional irreducible representation of ''S''<sub>3</sub>.

	~~==Characters==~~
	~~Any representation defines a [~~[~~Character theory\|character]] χ~~:''G'' → '''C'''. Such a function is constant on conjugacy classes of ''G'', a so-called [[class function]]; denote the ring of class functions by ''C''(''G''). The homomorphism ''R''(''G'') → ''C''(''G'') is injective, so that ''R''(''G'') can be identified with a subring of ''C''(''G''). For fields ''F'' whose characteristic divides the order of the group ''G'', the homomorphism from ''R''<sub>''F''</~~sub>(''G'') → ''C''(''G'') defined by [[modular representation theory\|Brauer characters]] is no longer injective.~~

	For a compact connected group ''R''(''G'') is isomorphic to the subring of ''R''(''T'') (where ''T'' is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961). For the general compact Lie group, see Segal (1968).

	~~==Adams operations==~~
	~~The ''Adams operations'' on the representation ring ''R''(''G'') are maps Ψ<sup>''k''<~~/~~sup> characterised by their effect on characters χ:~~
	~~:<math>\Psi^k \chi (g) = \chi(g^k) \~~ . ~~</math>~~

	~~The operations Ψ<sup>''k''</sup> are ring homomorphisms of ''R''(''G'') to itself, and on representations ρ of dimension ''d''~~

	~~:<math>\Psi^k (\rho) = N_k(\Lambda^1\rho,\Lambda^2\rho,\ldots,\Lambda^d\rho) \ </math>~~

	~~where the Λ<sup>''i''</sup>ρ are the [[exterior power]]s of ρ and ''N''<sub>''k''<~~/~~sub> is the ''k''-th [[Newton polynomial]].~~

	~~==References==~~
	*{{Citation \|authorlink1=Michael Atiyah \| last1=Atiyah \|first1= Michael F. \|last2=Hirzebruch\| first2=Friedrich \| title=Vector bundles and homogeneous spaces \|year=1961\|journal= Proc. Sympos. Pure Math. \| volume=III \| publisher=American Mathematical Society \| pages=7–38 \| mr=0139181 \| zbl=0108.17705 }}.
	*{{Citation \| last1=Bröcker\| first1=Theodor\| last2=tom Dieck \| first2=Tammo\| title=Representations of Compact Lie Groups \| publisher=[[Springer-Verlag]] \| location=New York, Berlin, Heidelberg, Tokyo \| series=[[Graduate Texts in Mathematics]] \| isbn= 0-387-13678-9\| mr = 1410059\| year=1985 \| volume=98 \| oclc=11210736 \| zbl=0581.22009 }}
	*{{Citation \|last=Segal \|first= Graeme \|title=The representation ring of a compact Lie group \|year=1968\|journal=Publ. Math. De l'IHES \| volume=34 \| pages=113–128 \| mr=0248277 \| zbl=0209.06203 }}.
	* {{citation \| title=Explicit Brauer Induction: With Applications to Algebra and Number Theory \| volume=40 \| series=Cambridge Studies in Advanced Mathematics \| first=V. P. \| last=Snaith \| authorlink= \| publisher=[[Cambridge University Press]] \| year=1994 \| isbn=0-521-46015-8 \| zbl=0991.20005 }}

	~~[[Category:Group theory]]~~
	~~[[Category:Finite groups]]~~
	~~[[Category:Lie groups]]~~
	~~[[Category:Representation theory of groups]~~]

en>Myasuda: avoid redirect

2013-07-18T04:05:58Z

avoid redirect

← Older revision		Revision as of 06:05, 18 July 2013
Line 1:		Line 1:
	~~<br><br>Gold has for long been~~ a ~~signifier~~ of ~~constant and efficient money. Individuals have been purchasing and investing in gold for hundreds~~ of ~~years~~. ~~Even these days~~, ~~gold continues to be~~ the ~~most secure and most profitable financial commitment chance~~ of ~~those around~~ the ~~world~~. ~~However~~, ~~today'~~s ~~gold rate has made people careful and put gold out~~ of ~~some people~~'~~s affordability~~. ~~Indian could possibly be the biggest gold customer~~ and it'~~s that isn~~'~~t going to change any time soon. Traders in India don~~'~~t see Gold as suspect~~, ~~just how many American investors do. This is due to~~ the ~~fact~~ of ~~India~~'~~s long connections to Gold that expand back thousands of years.~~<br><br>~~After you have~~ the ~~required know~~-~~how~~, ~~you can then select how you want to trade. There are two common techniques~~ of ~~to do it: spot~~ and ~~online gold trading~~. ~~Online gold trading offers certain advantages. The benefit~~ is ~~that you will never come~~ in ~~contact with the gold~~, ~~therefore you will not have~~ to ~~worry about its security or safety~~. ~~Furthermore in online gold trading~~, ~~you can use~~ the ~~available leverage offered by your brokerage firm. This allows you~~ [~~http:~~//~~trading~~.~~goldgrey~~.~~org~~/~~category~~/~~day-trading~~/ ~~wholesale] to buy much more gold with less bucks.~~<br><br>~~When you are looking~~ to ~~buy US Mint gold coins you have to be certain that you are getting~~ the ~~real deal~~. ~~Anyone can take what they believe to be a real coin and slap it on an online auction and sell it to~~ the ~~highest bidder. Perhaps you are getting~~ a ~~good deal but~~ the ~~authenticity of gold coin~~ is ~~still a question.~~<br><br>~~Upon reading about Sri Lanka we learned about the gold trade there~~ & ~~the stunning Ceylon sapphires~~. ~~It was decided that I would get my engagement~~ ring ~~made there. Whilst in Sri Lanka we had a day trip~~ to ~~the city of Kandy~~ & ~~the gold center there~~. ~~I got to design my own~~ ring ~~- a single Ceylon sapphire~~ of ~~1.25 carats set in 22 carat gold. It arrived at~~ the ~~hotel~~ the ~~day before the wedding!~~<br><br>~~Establishing forex trading isn~~'~~t a tricky approach. Sign up; deposit your to begin with trading "margin" amount and start buying and selling. It can not be easier than that.~~<br><br>~~Don~~'~~t forget that you can do this~~ for ~~all~~ the ~~other brackets WoW Twinks like~~ to ~~use~~, 20-~~29, 30~~-39, ~~forty~~-49. ~~You~~ can ~~probably tell that~~ the ~~record~~ of ~~objects that Twinks want~~ is ~~quite huge~~. ~~If you method perfectly, WoW Twinks are~~ a ~~excellent way~~ to ~~make some critical~~, ~~and I necessarily mean severe gold~~. ~~Superior luck~~.<br><br>~~So is there a bad time~~ to ~~buy gold bars for sale? Well~~, ~~only when you~~ are ~~directing money away from serious financial liabilities that require immediate attention~~. ~~But that is a very general piece~~ of ~~advice~~ - ~~never think~~ of ~~creating assets until you~~'~~ve eliminated your liabilities first~~. ~~So you see, the best time~~ to ~~invest~~ in ~~gold would be~~ - ~~as early as possible~~. ~~You better move fast!~~		In [[mathematics]], especially in the area of [[abstract algebra\|algebra]] known as [[representation theory]], the '''representation ring''' (or '''Green ring''' after [[J. A. Green]]) of a [[group (mathematics)\|group]] is a [[Ring (mathematics)\|ring]] formed from all the (isomorphism classes of the) finite-dimensional linear [[group representation\|representations]] of the group. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of [[algebraically closed field]]s of [[characteristic of a ring\|characteristic]] ''p'' where the [[Sylow subgroup\|Sylow ''p''-subgroups]] are [[cyclic group\|cyclic]] is also theoretically approachable.

			==Formal definition==
			Given a group ''G'' and a field ''F'', the elements of its '''representation ring''' ''R''<sub>''F''</sub>(''G'') are the formal differences of isomorphism classes of finite dimensional linear ''F''-representations of ''G''. For the ring structure, addition is given by the direct sum of representations, and multiplication by their [[tensor product]] over ''F''. When ''F'' is omitted from the notation, as in ''R''(''G''), then ''F'' is implicitly taken to be the field of complex numbers.

			==Examples==
			*For the complex representations of the [[cyclic group]] of order ''n'', the representation ring ''R''<sub>'''''C'''''</sub>(''C''<sub>''n''</sub>) is isomorphic to '''Z'''[''X'']/(''X''<sup>''n''</sup> − 1), where ''X'' corresponds to the complex representation sending a generator of the group to a primitive ''n''th root of unity.
			*More generally, the complex representation ring of a finite [[abelian group]] may be identified with the [[group ring]] of the [[character group]].
			*For the rational representations of the cyclic group of order 3, the representation ring ''R''<sub>'''''Q'''''</sub>(C<sub>3</sub>) is isomorphic to '''''Z'''''[''X'']/(''X''<sup>2</sup> − ''X'' − 2), where ''X'' corresponds to the irreducible rational representation of dimension 2.
			*For the modular representations of the cyclic group of order 3 over a field ''F'' of characteristic 3, the representation ring ''R''<sub>''F''</sub>(''C''<sub>3</sub>) is isomorphic to '''''Z'''''[''X'',''Y'']/(''X''<sup>2</sup> − ''Y'' − 1, ''XY'' − 2''Y'',''Y''<sup>2</sup> − 3''Y'').

			*The ring ''R''(S<sup>1</sup>) for the circle group is isomorphic to '''''Z'''''[''X'', ''X''<sup> −1</sup>]. The ring of real representations is the subring of ''R''(''G'') of elements fixed by the involution on ''R''(''G'') given by ''X'' → ''X''<sup> −1</sup>.

			*The ring ''R''<sub>'''''C'''''</sub>(''S''<sub>3</sub>) for the [[symmetric group]] on three points is isomorphic to '''Z'''[''X'',''Y'']/(''XY'' − ''Y'',''X''<sup>2</sup> − 1,''Y''<sup>2</sup> − ''X'' − ''Y'' − 1), where ''X'' is the 1-dimensional alternating representation and ''Y'' the 2-dimensional irreducible representation of ''S''<sub>3</sub>.

			==Characters==
			Any representation defines a [[Character theory\|character]] χ:''G'' → '''C'''. Such a function is constant on conjugacy classes of ''G'', a so-called [[class function]]; denote the ring of class functions by ''C''(''G''). The homomorphism ''R''(''G'') → ''C''(''G'') is injective, so that ''R''(''G'') can be identified with a subring of ''C''(''G''). For fields ''F'' whose characteristic divides the order of the group ''G'', the homomorphism from ''R''<sub>''F''</sub>(''G'') → ''C''(''G'') defined by [[modular representation theory\|Brauer characters]] is no longer injective.

			For a compact connected group ''R''(''G'') is isomorphic to the subring of ''R''(''T'') (where ''T'' is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961). For the general compact Lie group, see Segal (1968).

			==Adams operations==
			The ''Adams operations'' on the representation ring ''R''(''G'') are maps Ψ<sup>''k''</sup> characterised by their effect on characters χ:
			:<math>\Psi^k \chi (g) = \chi(g^k) \ . </math>

			The operations Ψ<sup>''k''</sup> are ring homomorphisms of ''R''(''G'') to itself, and on representations ρ of dimension ''d''

			:<math>\Psi^k (\rho) = N_k(\Lambda^1\rho,\Lambda^2\rho,\ldots,\Lambda^d\rho) \ </math>

			where the Λ<sup>''i''</sup>ρ are the [[exterior power]]s of ρ and ''N''<sub>''k''</sub> is the ''k''-th [[Newton polynomial]].

			==References==
			*{{Citation \|authorlink1=Michael Atiyah \| last1=Atiyah \|first1= Michael F. \|last2=Hirzebruch\| first2=Friedrich \| title=Vector bundles and homogeneous spaces \|year=1961\|journal= Proc. Sympos. Pure Math. \| volume=III \| publisher=American Mathematical Society \| pages=7–38 \| mr=0139181 \| zbl=0108.17705 }}.
			*{{Citation \| last1=Bröcker\| first1=Theodor\| last2=tom Dieck \| first2=Tammo\| title=Representations of Compact Lie Groups \| publisher=[[Springer-Verlag]] \| location=New York, Berlin, Heidelberg, Tokyo \| series=[[Graduate Texts in Mathematics]] \| isbn= 0-387-13678-9\| mr = 1410059\| year=1985 \| volume=98 \| oclc=11210736 \| zbl=0581.22009 }}
			*{{Citation \|last=Segal \|first= Graeme \|title=The representation ring of a compact Lie group \|year=1968\|journal=Publ. Math. De l'IHES \| volume=34 \| pages=113–128 \| mr=0248277 \| zbl=0209.06203 }}.
			* {{citation \| title=Explicit Brauer Induction: With Applications to Algebra and Number Theory \| volume=40 \| series=Cambridge Studies in Advanced Mathematics \| first=V. P. \| last=Snaith \| authorlink= \| publisher=[[Cambridge University Press]] \| year=1994 \| isbn=0-521-46015-8 \| zbl=0991.20005 }}

			[[Category:Group theory]]
			[[Category:Finite groups]]
			[[Category:Lie groups]]
			[[Category:Representation theory of groups]]

en>LokiClock at 15:28, 7 June 2012

2012-06-07T15:28:51Z

New page

Gold has for long been a signifier of constant and efficient money. Individuals have been purchasing and investing in gold for hundreds of years. Even these days, gold continues to be the most secure and most profitable financial commitment chance of those around the world. However, today's gold rate has made people careful and put gold out of some people's affordability. Indian could possibly be the biggest gold customer and it's that isn't going to change any time soon. Traders in India don't see Gold as suspect, just how many American investors do. This is due to the fact of India's long connections to Gold that expand back thousands of years. After you have the required know-how, you can then select how you want to trade. There are two common techniques of to do it: spot and online gold trading. Online gold trading offers certain advantages. The benefit is that you will never come in contact with the gold, therefore you will not have to worry about its security or safety. Furthermore in online gold trading, you can use the available leverage offered by your brokerage firm. This allows you [http://trading.goldgrey.org/category/day-trading/ wholesale] to buy much more gold with less bucks. When you are looking to buy US Mint gold coins you have to be certain that you are getting the real deal. Anyone can take what they believe to be a real coin and slap it on an online auction and sell it to the highest bidder. Perhaps you are getting a good deal but the authenticity of gold coin is still a question. Upon reading about Sri Lanka we learned about the gold trade there & the stunning Ceylon sapphires. It was decided that I would get my engagement ring made there. Whilst in Sri Lanka we had a day trip to the city of Kandy & the gold center there. I got to design my own ring - a single Ceylon sapphire of 1.25 carats set in 22 carat gold. It arrived at the hotel the day before the wedding! Establishing forex trading isn't a tricky approach. Sign up; deposit your to begin with trading "margin" amount and start buying and selling. It can not be easier than that. Don't forget that you can do this for all the other brackets WoW Twinks like to use, 20-29, 30-39, forty-49. You can probably tell that the record of objects that Twinks want is quite huge. If you method perfectly, WoW Twinks are a excellent way to make some critical, and I necessarily mean severe gold. Superior luck. So is there a bad time to buy gold bars for sale? Well, only when you are directing money away from serious financial liabilities that require immediate attention. But that is a very general piece of advice - never think of creating assets until you've eliminated your liabilities first. So you see, the best time to invest in gold would be - as early as possible. You better move fast!

← Older revision		Revision as of 00:40, 31 July 2014
Line 1:		Line 1:
	~~I am Alva and was born~~ on ~~3 February 1970~~. ~~My hobbies~~ are ~~Cheerleading~~ and ~~Rock stacking~~.<br><br>~~my site~~ ... [http://~~tinyurl~~.com/~~nrfwlnd nike air max~~]

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