Min-max theorem - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10667)

2015-01-04T07:58:16Z

WP:CHECKWIKI error fixes using AWB (10667)

← Older revision		Revision as of 09:58, 4 January 2015
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	~~Jerrie Swoboda~~ is ~~what you can call me and as well , I totally dig whom~~ name. ~~Managing people~~ is ~~my working day job now~~. As a ~~girl what Take into consideration like is to master croquet but I still can't make it my vocation really~~. ~~My husband's comments~~ and ~~I chose to maintain in~~ Massachusetts. Go to ~~my web-site to find out more~~: http://~~circuspartypanama~~.~~com~~<br><br>~~My web-site~~ [http://~~circuspartypanama~~.~~com Clash Of Clans Cheats Ipad Gems~~]		Some person who wrote some article is called Leland but it's not some of the most masucline name on the internet. To go to karaoke is the thing david loves most of . He works as a cashier. His wife and him live all through Massachusetts and he comes with everything that he [http://www.adobe.com/cfusion/search/index.cfm?term=&specifications&loc=en_us&siteSection=home specifications] there. He's not godd at design but locate want to check that website: http://prometeu.net<br><br>Also visit my blog; [http://prometeu.net clash of clans hack tool]

en>Kjetil1001: adding a section: counterexample in the non-Hermitian case

2014-02-11T20:30:03Z

adding a section: counterexample in the non-Hermitian case

← Older revision		Revision as of 22:30, 11 February 2014
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	~~In the [[mathematics\|mathematical]] field of [[representation theory]] a '''real representation'''~~ is ~~usually a [[group representation\|representation]] on a [[real number\|real]] [[vector space]] ''U'', but it~~ can ~~also mean a representation on a [[complex number\|complex]] vector space ''V'' with an invariant [[real structure]], i.e., an [[antilinear]] [[equivariant map]]~~		Jerrie Swoboda is what you can call me and as well , I totally dig whom name. Managing people is my working day job now. As a girl what Take into consideration like is to master croquet but I still can't make it my vocation really. My husband's comments and I chose to maintain in Massachusetts. Go to my web-site to find out more: http://circuspartypanama.com<br><br>My web-site [http://circuspartypanama.com Clash Of Clans Cheats Ipad Gems]
	~~:<math>j\colon V\to V\,</math>~~
	~~which satisfies~~
	~~:<math>j^2=+1.\,</math>~~
	The two viewpoints are equivalent because if ''U'' is a real vector space acted on by a group ''G'' (say), then ''V'' = ''U''&otimes;'''C''' is a representation on a complex vector space with an antilinear equivariant map given by [[complex conjugation]]. Conversely, if ''V'' is such a complex representation, ~~then ''U'' can be recovered as the [[fixed point set]] of ''j'' (the [[eigenspace]] with [[eigenvalue]] 1)~~.

	~~In [[physics]], where representations are often viewed concretely in terms of matrices, a real representation~~ is ~~one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors~~.

	~~A real representation on~~ a ~~complex vector space~~ is ~~isomorphic~~ to ~~its [[complex conjugate representation]],~~ but ~~the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a [[pseudoreal representation]]. An irreducible pseudoreal representation ''V~~'~~' is necessarily a [[quaternionic representation]]:~~ it ~~admits an invariant [[quaternionic structure]], i~~.~~e., an antilinear equivariant map~~
	~~:<math>j\colon V\~~to ~~V\,</math>~~
	~~which satisfies~~
	~~:<math>j^2=-1.\,</math>~~
	~~A [[direct sum of representations\|direct sum]] of real and quaternionic representations is neither real nor quaternionic~~ in ~~general~~.

	~~A representation on a complex vector space can also be isomorphic~~ to the [[dual representation]] of its complex conjugate. This happens precisely when the representation admits a nondegenerate invariant [[sesquilinear form]], e.g. a [[hermitian form]]. Such representations are sometimes said to ~~be complex or (pseudo-)hermitian.~~

	~~==Frobenius-Schur indicator==~~

	~~A criterion (for [[compact group]]s ''G'') for reality of irreducible representations in terms of [[character theory]] is based on the '''[[Frobenius-Schur indicator]]''' defined by~~
	:~~<math>\int_{g\in G}\chi(g^2)\,d\mu<~~/~~math>~~
	~~where ''χ'' is the character of the representation and ''μ'' is the [[Haar measure]] with μ(''G'') = 1. For a finite group, this is given by~~
	~~:<math>{1\over \|G\|}\sum_{g\in G}\chi(g^2).<~~/~~math>~~
	~~The indicator may take the values 1, 0 or −1. If the indicator is 1, then the representation is real~~. ~~If the indicator is zero, the representation is complex (hermitian),~~<~~ref~~>~~Any complex representation ''V'' of a compact group has an invariant hermitian form, so the significance of zero indicator is that there is no invariant nondegenerate complex bilinear form on ''V''.~~<~~/ref~~> ~~and if the indicator is −1, the representation is quaternionic.~~

	~~==Examples==~~

	~~All representation of the [[symmetric group]]s are real (and in fact rational), since we can build a complete set of [[irreducible representations]] using [[Young tableaux]].~~

	All representations of the [[special orthogonal group\|rotation group]]s are real, since they all appear as subrepresentations of [[tensor product]]s of copies of the fundamental representation, which is real.

	Further examples of real representations are the [[spinor]] representations of the [[spin group]]s in 8''k''−1, 8''k'', and 1 + 8''k'' dimensions for ''k'' = 1, 2, 3 ... . This periodicity ''[[Modular arithmetic\|modulo]]'' 8 is known in mathematics not only in the theory of [[Clifford algebra]]s, but also in [[algebraic topology]], in [[KO-~~theory]]; see~~ [~~[spin representation]]~~.

	~~==Notes==~~
	~~{{reflist\|1}}~~

	~~==References==~~
	*{{Fulton-Harris}}.
	*{{citation \|first=Jean-Pierre\|last= Serre \| title=Linear Representations of Finite Groups \| publisher=Springer-Verlag \| year=1977 \| isbn= 0-387-90190-6}}.

	~~[[Category:Representation theory]~~]

2620:101:F000:700:45C9:79C9:F19F:DA48: /* Proof */

2013-09-11T21:11:57Z

Proof

← Older revision		Revision as of 23:11, 11 September 2013
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	~~Most states wait until June~~, ~~July or even August to host their family backyard get togethers and festivities~~. ~~Here in Tucson~~, ~~Arizona~~, ~~though the weather great by mid-March for outdoor activities, and Tucsonans are ready begin grilling!~~<br><br>~~Your final option for creating privacy~~ on ~~your patio to~~ be ~~able to plant some tall growing plants inside~~ of ~~edges~~. ~~Ornamental grasses look nice with some varieties being known to grow up to thirteen feet~~ in ~~height~~. ~~They are also fast growing~~, which is ~~really~~ a ~~bonus should you want privacy fast~~. ~~hedges and medium height evergreens are also popular strategies for screening nosy neighbours from direct views of your patio location~~.<br><br><br><br>~~If you try~~ to ~~landscape while budgeting~~, ~~remember which you can complete~~ a ~~project in phases~~. ~~In fact, it is usually a good idea~~ to ~~break your project up into different steps and even seasons~~. ~~Is actually usually easier to achieve this~~ for ~~money. Write each part~~ in the ~~process down and select ones which are important to complete first.~~<br><br>~~Buying an at home also anyone to have insight as an integral part~~ of ~~a permanent community. On~~ the ~~other side hand, in~~ the ~~rented apartment or home~~, ~~one might feel temporary and less involved.~~<br><br>~~Owning your own home means gaining equity~~. If ~~ever~~ the ~~owner keeps~~ the ~~house long enough for it to rise above~~ the ~~initial cost~~ of the ~~purchase, then~~ that is ~~profit. This is one for this most essential and superb matters having home usage~~.<br>~~<br>If it becomes clear that your heating costs~~ is ~~usually a bit rrn excess of you to be able to be paying~~, ~~you will probably have someone install some better insulation to your dwelling~~. ~~While this~~ are ~~sometimes a bit pricey~~, it a ~~lot less than you would have to order inflated utility bills, over time~~.~~<br><br>There~~ are ~~two epidermis landscape gardening- formal and informal. If you find any formal arrangement~~, ~~just about be straight paths utilizing straightforward series. The entire ambiance appears~~ as ~~formal also as company~~. ~~You will obtain~~ the ~~reverse just in example~~ of ~~any informal understanding. You can always choose through~~ the ~~one~~ for the ~~two~~. ~~Option is yours~~.~~<br><br>If you have any type~~ of ~~questions pertaining to where and ways to use~~ [~~http~~:~~//www.hedgingplants.com/ hedge~~]~~, you could call us at the webpage.~~		In the [[mathematics\|mathematical]] field of [[representation theory]] a '''real representation''' is usually a [[group representation\|representation]] on a [[real number\|real]] [[vector space]] ''U'', but it can also mean a representation on a [[complex number\|complex]] vector space ''V'' with an invariant [[real structure]], i.e., an [[antilinear]] [[equivariant map]]
			:<math>j\colon V\to V\,</math>
			which satisfies
			:<math>j^2=+1.\,</math>
			The two viewpoints are equivalent because if ''U'' is a real vector space acted on by a group ''G'' (say), then ''V'' = ''U''&otimes;'''C''' is a representation on a complex vector space with an antilinear equivariant map given by [[complex conjugation]]. Conversely, if ''V'' is such a complex representation, then ''U'' can be recovered as the [[fixed point set]] of ''j'' (the [[eigenspace]] with [[eigenvalue]] 1).

			In [[physics]], where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors.

			A real representation on a complex vector space is isomorphic to its [[complex conjugate representation]], but the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a [[pseudoreal representation]]. An irreducible pseudoreal representation ''V'' is necessarily a [[quaternionic representation]]: it admits an invariant [[quaternionic structure]], i.e., an antilinear equivariant map
			:<math>j\colon V\to V\,</math>
			which satisfies
			:<math>j^2=-1.\,</math>
			A [[direct sum of representations\|direct sum]] of real and quaternionic representations is neither real nor quaternionic in general.

			A representation on a complex vector space can also be isomorphic to the [[dual representation]] of its complex conjugate. This happens precisely when the representation admits a nondegenerate invariant [[sesquilinear form]], e.g. a [[hermitian form]]. Such representations are sometimes said to be complex or (pseudo-)hermitian.

			==Frobenius-Schur indicator==

			A criterion (for [[compact group]]s ''G'') for reality of irreducible representations in terms of [[character theory]] is based on the '''[[Frobenius-Schur indicator]]''' defined by
			:<math>\int_{g\in G}\chi(g^2)\,d\mu</math>
			where ''χ'' is the character of the representation and ''μ'' is the [[Haar measure]] with μ(''G'') = 1. For a finite group, this is given by
			:<math>{1\over \|G\|}\sum_{g\in G}\chi(g^2).</math>
			The indicator may take the values 1, 0 or −1. If the indicator is 1, then the representation is real. If the indicator is zero, the representation is complex (hermitian),<ref>Any complex representation ''V'' of a compact group has an invariant hermitian form, so the significance of zero indicator is that there is no invariant nondegenerate complex bilinear form on ''V''.</ref> and if the indicator is −1, the representation is quaternionic.

			==Examples==

			All representation of the [[symmetric group]]s are real (and in fact rational), since we can build a complete set of [[irreducible representations]] using [[Young tableaux]].

			All representations of the [[special orthogonal group\|rotation group]]s are real, since they all appear as subrepresentations of [[tensor product]]s of copies of the fundamental representation, which is real.

			Further examples of real representations are the [[spinor]] representations of the [[spin group]]s in 8''k''−1, 8''k'', and 1 + 8''k'' dimensions for ''k'' = 1, 2, 3 ... . This periodicity ''[[Modular arithmetic\|modulo]]'' 8 is known in mathematics not only in the theory of [[Clifford algebra]]s, but also in [[algebraic topology]], in [[KO-theory]]; see [[spin representation]].

			==Notes==
			{{reflist\|1}}

			==References==
			*{{Fulton-Harris}}.
			*{{citation \|first=Jean-Pierre\|last= Serre \| title=Linear Representations of Finite Groups \| publisher=Springer-Verlag \| year=1977 \| isbn= 0-387-90190-6}}.

			[[Category:Representation theory]]

en>Xnn: /* Matrices */ exclude 0 from domain of R_A

2012-05-28T17:13:52Z

Matrices: exclude 0 from domain of R_A

New page

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