Datar–Mathews method for real option valuation: Difference between revisions

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In mathematics, '''Alvis–Curtis duality''' is a [[Duality (mathematics)|duality operation]] on the [[Character (mathematics)|characters]] of a [[reductive group]] over a [[finite field]], introduced by {{harvs|last=Curtis|first=Charles W.|authorlink=Charles W. Curtis|txt|year=1980}} and studied by his student {{harvs|last=Alvis|first=Dean|txt|year=1979}}. {{harvs|txt|last=Kawanaka|year1=1981|year2=1982}} introduced a similar duality operation for Lie algebras.
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Alvis–Curtis duality has order 2 and is an isometry on generalized characters.
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{{harvtxt|Carter|1985|loc=8.2}} discusses Alvis–Curtis duality in detail.
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==Definition==
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The dual ζ* of a character ζ of a finite group ''G'' with a split [[BN-pair]] is defined to be
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:<math>\zeta^*=\sum_{J\subseteq R}(-1)^J\zeta^G_{P_J}</math>
Here the sum is over all subsets ''J'' of the set ''R'' of simple roots of the Coxeter system of ''G''. The character ζ{{su|p=|b=''P''<sub>''J''</sub>}} is the '''truncation''' of ζ to the parabolic subgroup ''P''<sub>''J''</sub> of the subset ''J'', given by restricting ζ to ''P''<sub>''J''</sub> and then taking the space of invariants of the unipotent radical of ''P''<sub>''J''</sub>, and  ζ{{su|p=''G''|b=''P''<sub>''J''</sub>}} is the induced representation of ''G''. (The operation of  truncation is the adjoint functor of [[parabolic induction]].)
 
==Examples==
 
*The dual of the trivial character 1 is the [[Steinberg character]].
*{{harvtxt|Deligne|Lusztig|1983}} showed that the dual of a [[Deligne–Lusztig character]] ''R''{{su|b=T|p=θ}} is ε<sub>''G''</sub>ε<sub>''T''</sub>''R''{{su|b=T|p=θ}}.
*The dual of a [[cuspidal character]] χ is (–1)<sup>|Δ|</sup>χ, where Δ is the set of simple roots.
*The dual of the [[Gelfand–Graev character]] is the character taking value |''Z''<sup>''F''</sup>|''q''<sup>''l''</sup> on the regular unipotent elements and vanishing elsewhere.
 
==References==
 
*{{Citation | last1=Alvis | first1=Dean | title=The duality operation in the character ring of a finite Chevalley group | doi=10.1090/S0273-0979-1979-14690-1 | mr=546315 | year=1979 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=1 | issue=6 | pages=907–911}}
*{{Citation | last1=Carter | first1=Roger W. | author1-link=Roger Carter (mathematician) | title=Finite groups of Lie type. Conjugacy classes and complex characters.  | url=http://books.google.com/books?id=LvvuAAAAMAAJ | publisher=[[John Wiley & Sons]] | location=New York | series=Pure and Applied Mathematics (New York) | isbn=978-0-471-90554-7 | mr=794307 | year=1985}}
*{{Citation | last1=Curtis | first1=Charles W. | authorlink = Charles W. Curtis | title=Truncation and duality in the character ring of a finite group of Lie type | doi=10.1016/0021-8693(80)90185-4 | mr=563231 | year=1980 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=62 | issue=2 | pages=320–332}}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Lusztig | first2=George | title=Duality for representations of a reductive group over a finite field | doi=10.1016/0021-8693(82)90023-0 | mr=644236  | year=1982 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=74 | issue=1 | pages=284–291}}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Lusztig | first2=George | title=Duality for representations of a reductive group over a finite field. II | doi=10.1016/0021-8693(83)90202-8 | mr=700298  | year=1983 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=81 | issue=2 | pages=540–545}}
*{{Citation | last1=Kawanaka | first1=Noriaki | title=Fourier transforms of nilpotently supported invariant functions on a finite simple Lie algebra | url=http://projecteuclid.org/getRecord?id=euclid.pja/1195516260 | mr=637555 | year=1981 | journal=Japan Academy. Proceedings. Series A. Mathematical Sciences | issn=0386-2194 | volume=57 | issue=9 | pages=461–464}}
*{{Citation | last1=Kawanaka | first1=N. | title=Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field | doi=10.1007/BF01389363 | mr=679766 | year=1982 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=69 | issue=3 | pages=411–435}}
 
{{DEFAULTSORT:Alvis-Curtis duality}}
[[Category:Representation theory]]
[[Category:Duality theories]]

Latest revision as of 10:46, 5 May 2014

Google has been delivery Thomas More and more of its services to iOS all over the years, and a fresh one hardly launched today: users tooshie at once take on backrest movies and TV shows they've purchased from Google Sport on their iPhone, iPad, or iPod tinge. Design-wise, it's real redolent of the Google Dally network shop and is moderately standardised to the Humanoid app, only regrettably there are close to pretty good compromises in this newly app. For starters, you can't in reality steal movies or TV in the app itself; this isn't a surprise, as Orchard apple tree would be able to takings a issue of entirely those sales. 

Other Google Meet apps for iOS equal medicine and books also don't crack a store, and Amazon's apps likewise don't countenance you buy media. However, a larger lose is offline playback ?? there's no path to synchronize whatsoever subject matter for acting plump for when you don't stimulate an net connexion. That's a Major skip for a Mobile River video recording app and pretty a lot makes it only when utile when you bear a Wi-Fi connexion. You besides can't frolic on-the-go ?

? video volition alone current all over a Wi-Fi connective. At the selfsame least, videos soaked cursorily without whatsoever buffering and lineament looked solid, as substantially. If you've to a great extent invested with in the Google Work media ecosystem, it's respectable that there's instantly an pick to look on your videos on iOS ?? you toilet yet ship them to Chromecast if you neediness to view on your TV. 

That said, it does experience similar it serves a moderately little direct market, and without offline playback it's non in all likelihood to supercede Apple's have iTunes Fund for the huge legal age of users. Update: This berth originally stated that rented movies weren't usable on iOS, merely Google allow us recognize that both rented and purchased message should present up and be playable in the new app. The Google Meet Movies & TV app http://t.

co/7aKTxspxR8 is right away on iOS! Vex the outflank of Romp movies & TV shows on iOS & #Chromecast. ?? Google Bid (@GooglePlay) January 15, 2014

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