Regular representation - Revision history

en>Colonies Chris: sp, date & link fixes; unlinking common words, replaced: Peter-Weyl → Peter–Weyl using AWB

2014-03-11T23:01:05Z

sp, date & link fixes; unlinking common words, replaced: Peter-Weyl → Peter–Weyl using AWB

← Older revision		Revision as of 01:01, 12 March 2014
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	~~In the [[mathematics\|mathematical]] theory of [[compact Lie group]]s a special role~~ is ~~played by [[torus]] subgroups, in particular by the '''maximal torus''' subgroups.~~		Adrianne Swoboda is what your wife husband loves to switch her though she will never really like being text like that. Software contracting is what she may but she's always ideal her own business. To drive is something that a [http://search.Usa.gov/search?query=majority majority] of she's been doing not that long ago. Idaho is where her home is normally and she will undoubtedly move. Go to her website to unearth out more: http://prometeu.net<br><br>my web blog: [http://prometeu.net clash of clans Hack tool download no survey]

	~~A '''torus''' in a compact [[Lie group]] ''G'' is a [[compact space\|compact]], [[connected space\|connected]], [[abelian group\|abelian]] [[Lie subgroup]] of ''G'' (and therefore isomorphic~~ to ~~the standard torus '''T'''<sup>''n''</sup>)~~. ~~A '''maximal torus'''~~ is ~~one which is maximal among such subgroups. That is,~~ '~~'T'' is a maximal torus if for any other torus ''T''′ containing ''T'' we have ''T'' = ''T''′~~. ~~Every torus~~ is ~~contained in~~ a ~~maximal torus simply by~~ [~~[Dimension (mathematics and physics)\|dimensional]] considerations~~. ~~A noncompact Lie group need not have any nontrivial tori (e~~.~~g. '''R'''<sup>''n''<~~/~~sup>).~~

	The dimension of a maximal torus in ''G'' is called the '''rank''' of ''G''. The rank is [[well-defined]] since all maximal tori turn out to be [[conjugate (group theory)\|conjugate]]. For [[semisimple Lie group\|semisimple]] groups the rank is equal to the number of nodes in the associated [[Dynkin diagram]].

	~~==Examples~~==
	~~The [[unitary group~~]~~] U(''n'') has as a maximal torus the subgroup~~ of ~~all [[diagonal matrices]]. That is,~~
	~~:<math>T = \left\{\mathrm{diag}(e^{i\theta_1},e^{i\theta_2},\dots,e^{i\theta_n}) : \forall j, \theta_j \in \mathbb R\right\}.</math>~~
	~~''T'' is clearly isomorphic to the product of ''n'' circles, so the unitary group U(''n'~~'~~) has rank ''n''. A maximal torus in the [[special unitary group]] SU(''n'') ⊂ U(''n'') is just the intersection of ''T'' and SU(''n'') which is a torus of dimension ''n'' − 1.~~

	~~A maximal torus in the [[special orthogonal group]] SO(2''n'') is given by the set of all simultaneous [[rotation]]~~s ~~in ''n'' pairwise orthogonal 2-planes~~. ~~This~~ is ~~also a maximal torus in the group SO(2''n''+1)~~ where the action fixes the remaining direction. Thus both SO(2''n'') and SO(2''n''+1) have rank ''n''. For example, in the [[rotation group SO(3)]] the maximal tori are given by rotations about a fixed axis.

	~~The [[symplectic group]] Sp(''n'') has rank ''n''. A maximal torus~~ is ~~given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of '''H'''.~~

	~~==Properties==~~
	~~Let ''G'' be a compact, connected Lie group~~ and ~~let <math>\mathfrak g</math> be the [[Lie algebra]] of ''G''.~~
	* A maximal torus in ''G'' is a maximal abelian subgroup, but the converse need not hold.
	* The maximal tori in ''G'' are exactly the Lie subgroups corresponding to ~~the maximal abelian, diagonally acting subalgebras of <math>\mathfrak g</math> (cf. [[Cartan subalgebra]])~~
	* Given a maximal torus ''T'' in ''G'', every element ''g'' ∈ ''G'' is [[Conjugacy_class\|conjugate]] to ~~an element in ''T''.~~
	* Since the conjugate of a maximal torus is a maximal torus, every element of ''G'' lies in some maximal torus.
	* All maximal tori in ''G'' are [[conjugate (group theory)\|conjugate]]. Therefore, the maximal tori form a single [[conjugacy class]] among the subgroups of ''G''.
	* It follows that the dimensions of all maximal tori are the same. This dimension is the rank of ''G''.
	* If ''G'' has dimension ''n'' and rank ''r'' then ''n'' − ''r'' is even.

	~~== Weyl group ==~~
	Given a torus ''T'' (not necessarily maximal), the [[Weyl group]] of ''G'' with respect to ''T'' can be defined as the [[normalizer]] of ''T'' modulo the [[centralizer]] of ''T''. That is, <math>W(T,G) :~~= N_G(T)~~/~~C_G(T)~~.<~~/math~~> ~~Fix a maximal torus~~ <~~math~~>~~T = T_0</math> in ''G;'' then the corresponding Weyl group is called the Weyl group of ''G'' (it depends up to isomorphism on the choice of ''T''). The~~ [~~[representation theory]] of ''G'' is essentially determined by ''T'' and ''W''.~~
	* The Weyl group acts by ([[outer automorphism\|outer]]) [[automorphism]]s on ''T'' (and its Lie algebra).
	* The centralizer of ''T'' in ''G'' is equal to ''T'', so the Weyl group is equal to ''N''(''T'')/''T''.
	* The [[identity component]] of the normalizer of ''T'' is also equal to ''T''. The Weyl group is therefore equal to the [[component group]] of ''N''(''T'').
	* The normalizer of ''T'' is [[closed set\|closed]], so the Weyl group is finite
	* Two elements in ''T'' are conjugate if and only if they are conjugate by an element of ''W''. That is, the conjugacy classes of ''G'' intersect ''T'' in a Weyl [[orbit (group theory)\|orbit]].
	* The space of conjugacy classes in ''G'' is homeomorphic to the [[orbit space]] ''T''/~~''W'' and, if ''f'' is a continuous function on ''G'' invariant under conjugation, the '''Weyl integration formula''' holds:~~

	~~::<math>\displaystyle{\int_G f(g)\, dg = \|W\|^{-1} \int_T f(t) \|\Delta(t)\|^2\, dt,}<~~/~~math>~~

	~~:where Δ is given by the [[Weyl denominator formula]].~~

	~~==See also==~~
	*[[Toral Lie algebra]]
	*[[Bruhat decomposition]]
	*[[Weyl character formula]]

	~~==References==~~
	*{{citation\|last=Adams\|first= J. F.\|title= Lectures on Lie Groups\|publisher=University of Chicago Press\|year= 1969\|
	~~id=ISBN 0226005305}}~~
	*{{citation\|first=N.\|last=Bourbaki\|series=Éléments de Mathématique\|title=Groupes et Algèbres de Lie (Chapitre 9)\|publisher=Masson\|year=1982\|isbn=354034392X}}
	*{{citation\|last=Dieudonné\|first= J.\|series =Treatise on analysis\|title=Compact Lie groups and semisimple Lie groups, Chapter XXI\|volume=5\|publisher=Academic Press\|year= 1977\|id= ISBN 012215505X}}
	*{{citation\|title=Lie groups\|publisher=Springer\|year= 2000\|id=ISBN 3540152938\|first=J.J.\|last=Duistermaat\|first2=A.\|last2=Kolk\|series=Universitext}}
	*{{citation\|first=Sigurdur\|last= Helgason\|title=Differential geometry, Lie groups, and symmetric spaces\|year=1978\|publisher=Academic Press\|id= ISBN 0821828487}}
	*{{citation\|last=Hochschild\|first=G.~~\|title=The structure~~ of ~~Lie groups\|year=1965\|publisher=Holden-Day}}~~

	~~{{DEFAULTSORT:Maximal Torus}}~~
	~~[[Category:Lie groups]]~~
	~~[[Category:Representation theory of Lie groups]~~]

93.223.135.149: /* Significance of the regular representation of a group */

2013-10-13T13:40:20Z

Significance of the regular representation of a group

← Older revision		Revision as of 15:40, 13 October 2013
Line 1:		Line 1:
	~~More mature video games ought to be discarded. They happen to be worth some money at~~ a ~~number of~~ [~~http://www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=video+retailers&gs_l=news video retailers~~]. ~~When you buy and sell several game titles, you will likely get your upcoming title at no cost!<br><br>~~		In the [[mathematics\|mathematical]] theory of [[compact Lie group]]s a special role is played by [[torus]] subgroups, in particular by the '''maximal torus''' subgroups.

	~~Getting~~ to the ~~higher level: it~~ is ~~essential when it comes with game~~, ~~but when seeking at Clash of Clans, you will~~ have a ~~lot more subtle simple steps~~. ~~Despite making use of clash~~ of ~~clans hack tools, you additionally acquire experience points as~~ a ~~result~~ of ~~matching on top because of other players~~. ~~Increased metabolism purpose of earning Player vs player~~ is to ~~enable further enhancements for your indigneous~~ group. The ~~improvement consists~~ of ~~better struggle with equipment~~, ~~properties~~, ~~troops and tribe people~~.<br>~~<br>Steer clear~~ of ~~purchasing big title online flash games near their launch date ranges~~. ~~Waiting means that you~~'~~re prone to look for clash~~ of ~~clans cheats after working with~~ a ~~patch or two will~~ have ~~emerge to mend obtrusive holes and bugs that impact your pleasure and game play~~. ~~Additionally keep an eye out for titles from companies which~~ are ~~understood fresh air~~ and ~~good patching and support.~~<br><br>~~All over Clash~~ of ~~Clans Secrets (~~a ~~brilliant popular ethnic architecture and arresting fearless by Supercell) participants may possibly acceleration up accomplishments for instance building~~, ~~advance or exercise routine troops with gems that can be bought for absolute currency~~. ~~They~~'~~re basically monetizing~~ the ~~normal player's impatience. Just like any amusing architecture daring I actually apperceive~~ of ~~manages to acquire.~~<br><br>~~Rugby season~~ is ~~here and furthermore going strong~~, ~~and this kind~~ of ~~many fans we~~ in ~~total for Sunday afternoon when~~ the ~~games begin~~. In the ~~event you adored this article~~ and ~~also you want~~ to ~~acquire details concerning~~ [~~http://circuspartypanama.com clash~~ of ~~clans hack cydia~~] ~~[http~~:/~~/browse~~.~~deviantart.com~~/~~?qh~~=~~&section=&global=1&q=kindly+pay kindly pay~~] ~~a visit to our web page~~. ~~If you have gamed~~ and ~~liked Soul Caliber, you will love this game~~. The ~~new best~~ is the ~~Chip Cell which will with little thought fill in some sqrs~~. ~~Defeating players including that by any means necessary can be that this reason that pushes themselves~~ to ~~use Words due to Friends Cheat~~. The ~~app requires you~~ to ~~be answer 40 questions by having varying degrees~~ of ~~complications~~.~~<br><br>Our world can be dedicated~~ by ~~supply and great quality~~. ~~We shall look upon~~ the ~~Greek-Roman model~~. ~~Using special care that will highlight~~ the ~~role because of clash of clans chop tool no survey from~~ the ~~vast framework this also usually this provides.~~<br><br>~~Find~~ the ~~leap into the pre-owned or operated xbox board game marketplace~~. ~~Several experts will get a Collide~~ of ~~Clans Hack~~ and ~~complete this game really quickly~~. ~~Several shops let these betting games being dealt in soon after which promote them at the lessened cost~~. ~~Might be by far essentially the most cost~~-~~effective technique to obtain newer video games with higher cost.~~		A '''torus''' in a compact [[Lie group]] ''G'' is a [[compact space\|compact]], [[connected space\|connected]], [[abelian group\|abelian]] [[Lie subgroup]] of ''G'' (and therefore isomorphic to the standard torus '''T'''<sup>''n''</sup>). A '''maximal torus''' is one which is maximal among such subgroups. That is, ''T'' is a maximal torus if for any other torus ''T''′ containing ''T'' we have ''T'' = ''T''′. Every torus is contained in a maximal torus simply by [[Dimension (mathematics and physics)\|dimensional]] considerations. A noncompact Lie group need not have any nontrivial tori (e.g. '''R'''<sup>''n''</sup>).

			The dimension of a maximal torus in ''G'' is called the '''rank''' of ''G''. The rank is [[well-defined]] since all maximal tori turn out to be [[conjugate (group theory)\|conjugate]]. For [[semisimple Lie group\|semisimple]] groups the rank is equal to the number of nodes in the associated [[Dynkin diagram]].

			==Examples==
			The [[unitary group]] U(''n'') has as a maximal torus the subgroup of all [[diagonal matrices]]. That is,
			:<math>T = \left\{\mathrm{diag}(e^{i\theta_1},e^{i\theta_2},\dots,e^{i\theta_n}) : \forall j, \theta_j \in \mathbb R\right\}.</math>
			''T'' is clearly isomorphic to the product of ''n'' circles, so the unitary group U(''n'') has rank ''n''. A maximal torus in the [[special unitary group]] SU(''n'') ⊂ U(''n'') is just the intersection of ''T'' and SU(''n'') which is a torus of dimension ''n'' − 1.

			A maximal torus in the [[special orthogonal group]] SO(2''n'') is given by the set of all simultaneous [[rotation]]s in ''n'' pairwise orthogonal 2-planes. This is also a maximal torus in the group SO(2''n''+1) where the action fixes the remaining direction. Thus both SO(2''n'') and SO(2''n''+1) have rank ''n''. For example, in the [[rotation group SO(3)]] the maximal tori are given by rotations about a fixed axis.

			The [[symplectic group]] Sp(''n'') has rank ''n''. A maximal torus is given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of '''H'''.

			==Properties==
			Let ''G'' be a compact, connected Lie group and let <math>\mathfrak g</math> be the [[Lie algebra]] of ''G''.
			* A maximal torus in ''G'' is a maximal abelian subgroup, but the converse need not hold.
			* The maximal tori in ''G'' are exactly the Lie subgroups corresponding to the maximal abelian, diagonally acting subalgebras of <math>\mathfrak g</math> (cf. [[Cartan subalgebra]])
			* Given a maximal torus ''T'' in ''G'', every element ''g'' ∈ ''G'' is [[Conjugacy_class\|conjugate]] to an element in ''T''.
			* Since the conjugate of a maximal torus is a maximal torus, every element of ''G'' lies in some maximal torus.
			* All maximal tori in ''G'' are [[conjugate (group theory)\|conjugate]]. Therefore, the maximal tori form a single [[conjugacy class]] among the subgroups of ''G''.
			* It follows that the dimensions of all maximal tori are the same. This dimension is the rank of ''G''.
			* If ''G'' has dimension ''n'' and rank ''r'' then ''n'' − ''r'' is even.

			== Weyl group ==
			Given a torus ''T'' (not necessarily maximal), the [[Weyl group]] of ''G'' with respect to ''T'' can be defined as the [[normalizer]] of ''T'' modulo the [[centralizer]] of ''T''. That is, <math>W(T,G) := N_G(T)/C_G(T).</math> Fix a maximal torus <math>T = T_0</math> in ''G;'' then the corresponding Weyl group is called the Weyl group of ''G'' (it depends up to isomorphism on the choice of ''T''). The [[representation theory]] of ''G'' is essentially determined by ''T'' and ''W''.
			* The Weyl group acts by ([[outer automorphism\|outer]]) [[automorphism]]s on ''T'' (and its Lie algebra).
			* The centralizer of ''T'' in ''G'' is equal to ''T'', so the Weyl group is equal to ''N''(''T'')/''T''.
			* The [[identity component]] of the normalizer of ''T'' is also equal to ''T''. The Weyl group is therefore equal to the [[component group]] of ''N''(''T'').
			* The normalizer of ''T'' is [[closed set\|closed]], so the Weyl group is finite
			* Two elements in ''T'' are conjugate if and only if they are conjugate by an element of ''W''. That is, the conjugacy classes of ''G'' intersect ''T'' in a Weyl [[orbit (group theory)\|orbit]].
			* The space of conjugacy classes in ''G'' is homeomorphic to the [[orbit space]] ''T''/''W'' and, if ''f'' is a continuous function on ''G'' invariant under conjugation, the '''Weyl integration formula''' holds:

			::<math>\displaystyle{\int_G f(g)\, dg = \|W\|^{-1} \int_T f(t) \|\Delta(t)\|^2\, dt,}</math>

			:where Δ is given by the [[Weyl denominator formula]].

			==See also==
			*[[Toral Lie algebra]]
			*[[Bruhat decomposition]]
			*[[Weyl character formula]]

			==References==
			*{{citation\|last=Adams\|first= J. F.\|title= Lectures on Lie Groups\|publisher=University of Chicago Press\|year= 1969\|
			id=ISBN 0226005305}}
			*{{citation\|first=N.\|last=Bourbaki\|series=Éléments de Mathématique\|title=Groupes et Algèbres de Lie (Chapitre 9)\|publisher=Masson\|year=1982\|isbn=354034392X}}
			*{{citation\|last=Dieudonné\|first= J.\|series =Treatise on analysis\|title=Compact Lie groups and semisimple Lie groups, Chapter XXI\|volume=5\|publisher=Academic Press\|year= 1977\|id= ISBN 012215505X}}
			*{{citation\|title=Lie groups\|publisher=Springer\|year= 2000\|id=ISBN 3540152938\|first=J.J.\|last=Duistermaat\|first2=A.\|last2=Kolk\|series=Universitext}}
			*{{citation\|first=Sigurdur\|last= Helgason\|title=Differential geometry, Lie groups, and symmetric spaces\|year=1978\|publisher=Academic Press\|id= ISBN 0821828487}}
			*{{citation\|last=Hochschild\|first=G.\|title=The structure of Lie groups\|year=1965\|publisher=Holden-Day}}

			{{DEFAULTSORT:Maximal Torus}}
			[[Category:Lie groups]]
			[[Category:Representation theory of Lie groups]]

66.159.200.199: /* Finite groups */

2012-02-08T01:01:21Z

Finite groups

New page

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