Vandermonde matrix - Revision history

143.215.124.243: Vandermonde determinants are important to people in a great many contexts - the way this fact was stated would needlessly worry people who are not familiar with abstract algebra.

2014-11-24T20:51:40Z

Vandermonde determinants are important to people in a great many contexts - the way this fact was stated would needlessly worry people who are not familiar with abstract algebra.

← Older revision		Revision as of 22:51, 24 November 2014
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en>Marc van Leeuwen: /* Properties */ Added restriction to square matrices where it was obviously necessary

2014-03-05T06:12:54Z

Properties: Added restriction to square matrices where it was obviously necessary

← Older revision		Revision as of 08:12, 5 March 2014
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	In the [[mathematics\|mathematical]] field of [[representation theory]], a '''projective representation''' of a [[group (mathematics)\|group]] ''G'' on a [[vector space]] ''V'' over a [[field (mathematics)\|field]] ''F'' is a [[group homomorphism]] from ''G'' to ~~the [[projective linear group]]~~		Pleased to meet you! My husband and my name is Eusebio in addition , I think it music quite good when say it. My house is now in Vermont and I don't wish on changing it. [https://www.Flickr.com/search/?q=Software+encouraging Software encouraging] has been my 24-hour period job for a while. To farrenheit is the only sport my wife doesn't approve of. I'm not good at web development but you might want to check my website: http://prometeu.net<br><br>my web-site [http://prometeu.net clash of clans hack android]
	~~:PGL(''V'',''F'') = GL(''V'',''F'')/''F''<sup>&lowast;</sup>~~
	~~where GL(''V'',''F'') is the [[general linear group]] of invertible linear transformations of ''V'' over ''F''~~ and ~~''F''<sup>*</sup> here~~ is ~~the [[normal subgroup]] consisting of multiplications of vectors~~ in ~~''V'' by nonzero elements of ''F'' (that is~~, ~~scalar multiples of the identity; [[scalar transformation]]s).<ref>{{Harvnb\|Gannon\|2006\|pp=176–179}}.</ref>~~

	~~==Linear representations and projective representations==~~

	~~One way in which a projective representation can arise is by taking a linear [[group representation]] of ''G'' on ''V'' and applying the quotient map~~

	~~:<math>GL(V,F) \rightarrow PGL(V, F)</math>~~

	which is the quotient by the subgroup ''F''<sup>∗</sup> of [[scalar transformation]]s ([[diagonal matrices]] with all diagonal entries equal). The interest for algebra is in the process in the other direction: given a ''projective representation'', try to 'lift' it ~~to a conventional ''linear representation''.~~

	~~[[File:Projective-representation-lifting.svg\|225px\|thumb\|A projective representation of ''G'' can be pulled back to a linear representation of a central extension ''C'' of ''G.'']]~~
	~~In general, given a projective representation {{math\|''ρ'': ''G'' → PGL(''V'')}}~~ it ~~cannot be lifted to a linear representation {{math\|''G'' → GL(''V'')}}, and the [[obstruction theory\|obstruction]] to this lifting can be understood via group homology, as described below~~. However, one ''can'' lift a projective representation of ''G'' to a linear representation of a different group ''C,'' which will be a [[central extension (mathematics)\|central extension]] of ''G.'' To understand this, note that GL(''V'') → PGL(''V'') is a central extension of PGL, meaning that the kernel is ~~central (~~in fact, is exactly the center of GL). One can [[pullback\|pull back]] the projective representation {{math\|''ρ'': ''G'' → PGL(''V'')}} along the quotient map, obtaining a ''linear'' representation {{math\|''σ'': ''C'' → GL(''V'')}} and '~~'C'' will be a central extension of ''G'' because~~ it ~~is a pullback of a central extension. Thus projective representations of ''G'' can be understood in terms of linear representations of (certain) central extensions of ''G~~.~~'' Notably, for ''G'' a [[perfect group]] there is a single [~~[~~universal perfect central extension]] of ''G'' that can be used.~~

	~~===Group cohomology===~~
	The analysis of the lifting question involves [[group cohomology]]. Indeed, if one introduces for ''g'' in ''G'' a lifted element ''L''(''g'') in lifting from PGL(''V'') back to GL(''V''), the lifts must satisfy

	:~~<math>L(gh) = c(g,h)L(g)L(h)</math>~~

	~~for some scalar ''c''(''g'',''h'') in ''F''<sup>∗<~~/~~sup>. The 2-cocycle or [[Schur multiplier]] ''c'' must satisfy the cocycle equation~~

	~~:<math> c(h,k)c(g,hk)= c(g,h) c(gh,k)<~~/~~math>~~

	~~for all ''g'', ''h'', ''k'' in ''G''~~. ~~This ''c'' depends on the choice of the lift ''L'', but a different choice of lift ''L′''(''g'')= ''f''(''g'') ''L''(''g'') will result in a new cocycle~~

	~~:<math>c^\prime(g,h) = f(gh)f(g)^{-1} f(h)^{-1} c(g,h)</math>~~

	~~cohomologous to ''c''~~. ~~Thus ''L'' defines a unique class in H<sup>2<~~/~~sup>(''G'', ''F''<sup>∗<~~/~~sup>), which need not be trivial. For example, in the case of the [[symmetric group~~]~~] and [[alternating group]], Schur proved that there is exactly one non~~-~~trivial class of Schur multiplier and completely determined all the corresponding irreducible representations.<ref>{{harvnb\|Schur\|1911}}</ref>~~

	~~It is shown, however, that this leads to an [[extension problem]]~~ for ~~''G''. If ''G'' is correctly extended we can speak of~~ a ~~linear representation of the extended group, which gives back the initial projective representation on factoring by ''F''<sup>∗</sup> and the extending subgroup~~. ~~The solution~~ is ~~always a [[Group extension#Central extension\|central extension]]. From [[Schur's lemma]], it follows that~~ the ~~[[irreducible representation]]s of central extensions of ''G'~~'~~, and the irreducible projective representations~~ of ~~''G'', describe essentially the same questions of representation theory~~.

	~~==Projective representations of Lie groups==~~
	~~{{see also\|Spinor}}~~
	Studying projective representations of [[Lie group]]s leads one to consider true representations of their central extensions (see [[Group extension#Lie groups]]). In many cases of interest it suffices to consider representations of [[covering group]]s; for a connected Lie group '~~'G'', this amounts~~ to ~~studying the representations of the Lie algebra of ''G''. Notable cases of covering groups giving interesting projective representations~~:

	* The [[special orthogonal group]] SO(''n'',''F'') is doubly covered by the [[Spin group]] Spin(''n'',''F''). In particular, the [[rotation group SO(3)\|group SO(3,'''R''')]] (the rotation group in 3 dimensions) is doubly covered by [[special unitary group\|SU(2)]]. This has important applications in quantum mechanics, as the [[representation theory of SU(2)\|study of representations of SU(2)]] leads to a [[Galilean relativity\|low-velocity]] theory of [[spin (physics)\|spin]].
	* The group [[Lorentz group\|SO<sup>+</~~sup>(3;1)]], isomorphic to the [[Möbius group]], is likewise doubly covered by [[special linear group\|SL<sub>2<~~/~~sub>]]('''C'''). Both are supergroups of aforementioned SO(3) and SU(2) respectively and form a [[special relativity\|relativistic]] spin theory~~.
	* The [[orthogonal group]] O(''n'') is double covered by the [[Pin group]] Pin<~~sub~~>±<~~/sub~~>~~(''n'').~~
	* The [[symplectic group]] Sp(2''n'') is double covered by the [[metaplectic group]] Mp(2''n'').

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*{{citation\|first=I.\|last=Schur\|authorlink=Issai Schur\|title=Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen\|year=1911\|journal=[~~[Crelle's Journal]]\|pages=155–250\|volume=139\|url=~~http://~~gdz.sub.uni-goettingen~~.~~de/no_cache/en/dms/load/img/?IDDOC=261150~~
	}}
	*{{citation\|title=Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics\|first=Terry\|last= Gannon\|publisher=Cambridge University Press\|year= 2006\|isbn=978-0-521-83531-2}}

	~~==See also==~~
	*[[Affine representation]]
	*[[Group action]]

	~~[[Category:Homological algebra]]~~
	~~[[Category:Group theory]]~~
	~~[[Category:Representation theory]]~~
	~~[[Category:Representation theory~~ of ~~groups]~~]

en>BattyBot: fixed CS1 errors: dates & General fixes using AWB (9832)

2014-01-01T23:55:11Z

fixed CS1 errors: dates & General fixes using AWB (9832)

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en>Melchoir: /* References */ +Further reading, preprint

2012-04-23T01:28:59Z

References: +Further reading, preprint

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