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In mathematics, specifically in the [[representation theory]] of [[group (mathematics)|group]]s and [[algebra over a field|algebras]], an '''irreducible representation''' or '''irrep''' of an algebraic structure is a nonzero representation that has no proper closed subrepresentations.
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Every finite dimensional [[unitary representation]] on a [[Hermitian]] vector space {{math|''V''}} is the [[direct sum]] of irreducible representations. As irreducible representations are always '''indecomposable''' (i.e. cannot be decomposed further into a direct sum of representations), these terms are often confused; however, in general there are many reducible but indecomposable representations, such as the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices.
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Group representation theory was generalized by [[Richard Brauer]] from the 1940s to give [[modular representation theory]], in which the matrix operators act over a [[field (mathematics)|field]] {{math|''K''}} of arbitrary [[characteristic (algebra)|characteristic]], rather than a vector of [[real number|real]] or [[complex number]]s.  The structure analogous to an irreducible representation in the resulting theory is a [[simple module]].{{citation needed|date=July 2013}}
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{{details|Group representation}}
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Let {{math|''ρ''}} be a representation {{math|''ρ'': ''G'' → GL(''V'')}} of a group {{math|''G''}} where {{math|''V''}} is a [[vector space]] over a [[field (mathematics)|field]] {{math|''F''}}. If we pick a basis {{math|'''B'''}} for {{math|''V''}}, {{math|''ρ''}} can be thought of as a function (a [[homomorphism]]) from a group onto a set of invertible matrices and in this context is called a '''matrix representation'''. However, it simplifies things greatly if we think of the space {{math|''V''}} without a basis.
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A [[linear subspace]] {{math|''W'' ⊂ ''V''}} is called '''''G''-invariant''' if {{math|''gw'' ∈ ''W''}} for all {{math|''g'' ∈ ''G''}} and all {{math|''w'' ∈ ''W''}}. The [[Restriction (mathematics)|restriction]] of {{math|''ρ''}} to a {{math|''G''}}-invariant subspace {{math|''W'' ⊂ ''V''}} is known as a '''subrepresentation'''. A representation {{math|''ρ'': ''G'' → GL(''V'')}} is said to be '''irreducible''' if it has only [[trivial (mathematics)|trivial]] subrepresentations (all representations can form a subrepresentation with the trivial ''G''-invariant subspaces, e.g. the whole vector space {{math|''V''}}, and [[zero vector space|{0}]]). If there is a proper non-trivial invariant subspace, {{math|''ρ''}} is said to be '''reducible'''.
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===Notation and terminology of group representations===
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Group elements can be represented by [[matrix (mathematics)|matrices]], although the term "represented" has a specific and precise meaning in this context. A representation of a group is a mapping from the group elements to the [[general linear group]] of matrices. As notation, let {{math|''a'', ''b'', ''c''...}} denote elements of a group {{math|''G''}} with group product signified without any symbol, so {{math|''ab''}} is the group product of {{math|''a''}} and {{math|''b''}} and is also an element of {{math|''G''}}, and let representations be indicated by {{math|''D''}}. The '''representation of ''a''''' is written
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:<math>D(a) = \begin{pmatrix}
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D(a)_{11} & D(a)_{12} & \cdots & D(a)_{1n} \\
 
D(a)_{21} & D(a)_{22} & \cdots & D(a)_{2n} \\
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\vdots & \vdots & \ddots & \vdots \\
 
D(a)_{n1} & D(a)_{n2} & \cdots & D(a)_{nn} \\
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By definition of group representations, the representation of a group product is translated into [[matrix multiplication]] of the representations:
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:<math>D(ab) = D(a)D(b) </math>
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If {{math|''e''}} is the [[identity element]] of the group (so that {{math|''ae'' {{=}} ''ea'' {{=}} ''a''}}, etc.), then {{math|''D''(''e'')}} is an [[identity matrix]], or identically a block matrix of identity matrices, since we must have
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:<math>D(ea) = D(ae) = D(a)D(e) = D(e)D(a) = D(a) </math>
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and similarly for all other group elements.
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A representation is decomposable if a [[Matrix similarity|similar matrix]] {{math|''P''}} can be found for the [[matrix similarity|similarity transformation]]:<ref name="Wigner p 73">{{cite book| author=E.P. Wigner|title=Group theory and its application to the quantum mechanics of atomic spectra|year=1959|series=Pure and applied physics|page=73|publisher=Academic press|isbn=}}</ref>
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:<math> D(a) \rightarrow P^{-1} D(a) P</math>
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which [[matrix diagonalization|diagonalizes]] every matrix in the representation into the same pattern of [[diagonal matrix|diagonal]] [[block matrix|block]]s – each of the blocks are representation of the group independent of each other. The representations {{math|''D''(''a'')}} and {{math|''P''<sup>−1</sup>''D''(''a'')''P''}} are said to be '''equivalent representations'''.<ref>{{cite book|author= W.K. Tung|title=Group Theory in Physics|page=32|publisher=World Scientific|year=1985|url=http://books.google.co.uk/books?id=O89tgpOBO04C&printsec=frontcover&dq=group+theory+in+physics&hl=en&sa=X&ei=Xsd-UdmjONKg0wW96ICwBg&redir_esc=y#v=onepage&q=group%20theory%20in%20physics&f=false|isbn=997-1966-565}}</ref> The representation can be decomposed into a [[direct sum of matrices|direct sum of ''k'' matrices]]:
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:<math>D(a) = \begin{pmatrix}
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D^{(1)}(a) & 0 & \cdots & 0 \\
 
0 & D^{(2)}(a) & \cdots & 0 \\
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0 & 0 & \cdots & D^{(k)}(a) \\
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\end{pmatrix} = D^{(1)}(a) \oplus D^{(2)}(a) \oplus \cdots \oplus D^{(k)}(a) </math>
 
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so {{math|''D''(''a'')}} is '''decomposable''', and it is customary to label the decomposed matrices by a superscript in brackets, as in {{math|''D''<sup>(''n'')</sup>(''a'')}} for {{math|''n'' {{=}} 1, 2, ..., ''k''}}, although some authors just write the numerical label without brackets.
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The dimension of {{math|''D''(''a'')}} is the sum of the dimensions of the blocks:
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:<math> \mathrm{dim}[D(a)] = \mathrm{dim}[D^{(1)}(a)] + \mathrm{dim}[D^{(2)}(a)] + \ldots + \mathrm{dim}[D^{(k)}(a)] </math>
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If this is not possible, then the representation is indecomposable.<ref name="Wigner p 73"/><ref name="Tung 33">{{cite book|author= W.K. Tung|title=Group Theory in Physics
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{{see also|Symmetry in quantum mechanics|Jahn–Teller effect}}
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In [[quantum physics]] and [[quantum chemistry]], each set of [[Degenerate energy levels|degenerate eigenstates]] of the [[Hamiltonian operator]] makes up a representation of the symmetry group of the Hamiltonian, that barring accidental degeneracies will correspond to an irreducible representation. Identifying the irreducible representations therefore allows one to label the states, predict how they will [[energy level splitting|split]] under perturbations; and predict non-zero transition elements.
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In quantum mechanics, irreducible representations of the symmetry group of the system label the energy levels of the system, allowing the [[selection rule]]s to be determined.<ref>{{cite web|publisher=Oxford Dictionary of Chemistry|title=A Dictionary of Chemistry, Answers.com|edition=6th|url=http://www.answers.com/topic/irreducible-representation}}</ref>
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{{main|Representation theory of Lie groups}}
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The irreps of {{math|''D''('''K''')}} and {{math|''D''('''J''')}}, where {{math|'''J'''}} is the generator of rotations and {{math|'''K'''}} the generator of boosts, can be used to build to spin representations of the Lorentz group, because they are related to the spin matrices of quantum mechanics. This allows them to derive [[relativistic wave equation]]s.<ref>{{cite news
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| author = T. Jaroszewicz, P.S Kurzepa
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===Associative algebras===
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*[[Simple module]]
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*[[Indecomposable module]]
 
*[[Representation of an associative algebra]]
  <li>[http://general.assembly.codesria.org/spip.php?article87&lang=pt/&quot;/ http://general.assembly.codesria.org/spip.php?article87&lang=pt/&quot;/]</li>
 
 
===Lie groups===
  <li>[http://lab.nqnwebs.com/lavoz_bak/spip.php?article13211/ http://lab.nqnwebs.com/lavoz_bak/spip.php?article13211/]</li>
 
 
* [[Representation theory of Lie algebras]]
  <li>[http://www.reo8.moe.go.th/index.php/forum/9/316526-replica-ray-ban-aviators-8-samsung-smartphones-price-dropteletalk-3g-modem#316526 http://www.reo8.moe.go.th/index.php/forum/9/316526-replica-ray-ban-aviators-8-samsung-smartphones-price-dropteletalk-3g-modem#316526]</li>
* [[Representation theory of SU(2)]]
 
* [[Representation theory of SL2(R)]]
</ul>
* [[Representation theory of the Galilean group]]
* [[Representation theory of diffeomorphism groups]]
* [[Representation theory of the Poincaré group]]
 
==References==
 
{{reflist}}
 
===Books===
 
*{{cite book|author=[[Hermann Weyl|H. Weyl]]|title=The theory of groups and quantum mechanics|page=203|publisher=Courier Dover Publications|year=1950|url=http://books.google.co.uk/books?id=jQbEcDDqGb8C&pg=PA203&dq=magnetic+moments+in+relativistic+quantum+mechanics&hl=en&sa=X&ei=X1h4Uc79Dcb40gWI1YDwCg&redir_esc=y#v=onepage&q=magnetic%20moments%20in%20relativistic%20quantum%20mechanics&f=false}}
 
*{{cite book|author=A. D. Boardman, D. E. O'Conner, P. A. Young|title=Symmetry and its applications in science|page=|publisher=McGraw Hill|year=1973|isbn=0-07-084011-3}}
 
*{{cite book|author=V. Heine|title=Group theory in quantum mechanics: an introduction to its present usage|page=|publisher=Dover|year=republished: 2007 original: 1960|url=http://archive.org/details/GroupTheoryInQuantumMechanics|isbn=0-07-084011-3}}
 
*{{cite book|author=V. Heine|title=Group Theory in Quantum Mechanics: An Introduction to Its Present Usage
|page=|publisher=Courier Dover Publications|year=1993|url=http://books.google.co.uk/books?id=NayFD34uEu0C&pg=PA363&dq=lorentz+group+in+relativistic+quantum+mechanics&hl=en&sa=X&ei=8MZ-Ua-uNqaK0AX57YGYCA&ved=0CEAQ6AEwAQ#v=onepage&q=lorentz%20group%20in%20relativistic%20quantum%20mechanics&f=false|isbn=048-6675-858}}
 
*{{cite book|title=Quantum Mechanics|author=E. Abers|publisher=Addison Wesley|year=2004|page=425|isbn=978-0-13-146100-0}}
 
*{{cite book| author=B. R. Martin, G.Shaw|title=Particle Physics|edition=3rd|publisher=Manchester Physics Series, John Wiley & Sons|pages=3|isbn=978-0-470-03294-7}}
 
*{{citation
|last = Weinberg
|first = S
|year = 1995
|title = The Quantum Theory of Fields
|pages=230–231
|volume = 1
|publisher=Cambridge university press
|isbn = 0-521-55001-7
}}
*{{citation
| last = Weinberg
| first = S
| year = 1996
| title = The Quantum Theory of Fields
| volume = 2
|publisher=Cambridge university press
|isbn = 0-521-55002-7
}}
*{{citation
| last = Weinberg
| first = S
| year = 2000
| title = The Quantum Theory of Fields
| volume = 3
|publisher=Cambridge university press
|isbn = 0-521-66000-9
}}
*{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books|page=| year=2007 | isbn=0-679-77631-1}}
 
*{{cite book|title=Molecular Quantum Mechanics (Parts 1 and 2): An introduction to quantum chemistry|volume=1|pages=125–126|author=P. W. Atkins|publisher=Oxford University Press|year=1970|isbn=0-19-855129-0}}
 
===Papers===
 
*{{cite journal|author1=Bargmann, V.|author2=Wigner, E. P.|title=Group theoretical discussion of relativistic wave equations|year=1948|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=34|pages=211–23|url=http://www.pnas.org/cgi/content/citation/34/5/211|issue=5}}
*{{cite journal
| author = E. Wigner
| year = 1937
| title = On Unitary Representations Of The Inhomogeneous Lorentz Group
| journal = Annals of Mathematics
| location =
| volume = 40
| number = 1
| page = 149
| publisher =
| url = http://courses.theophys.kth.se/SI2390/wigner_1939.pdf
}}
 
==Further reading==
 
*{{cite web|last=Artin|first=Michael|title=Noncommutative Rings|url=http://math.mit.edu/~etingof/artinnotes.pdf|year=1999|location=Chapter V}}
 
==External links==
 
*[http://www.crystallography.fr/mathcryst/pdf/nancy2010/Aroyo_reps2010.pdf (2010) ''Commission on Mathematical and Theoretical Crystallography, Summer Schools on Mathematical Crystallography'']
*[http://cft.fis.uc.pt/eef/evbgroups.pdf ''Some notes on group theory'']
*[http://math.berkeley.edu/~teleman/math/RepThry.pdf ''Representation Theory'']
*[http://panda.unm.edu/Courses/Finley/p467/handouts/YoungTableauxSubs.pdf ''Some Notes on Young Tableaux as useful for irreps of su(n)'']
*[http://www.huntresearchgroup.org.uk/teaching/teaching_MOs_year2/L2_Addn_Symm_Labels.pdf ''Irreducible Representation (IR) Symmetry Labels'']
*[http://www.physics.indiana.edu/~dermisek/QFT_08/qft-I-14-4p.pdf ''Representations of Lorentz Group'']
*[http://einrichtungen.ph.tum.de/T30f/lec/QFT/groups.pdf ''Representations of Lorentz and Poincaré groups'']
*[http://www.math.columbia.edu/~woit/QM/lorentzrep.pdf ''Quantum Mechanics for Mathematicians: Representations of the Lorentz Group'']
*[http://pages.cs.wisc.edu/~guild/symmetrycompsproject.pdf ''Representations of the Symmetry Group of Spacetime'']
*[http://panda.unm.edu/Courses/Finley/P495/handouts/PoincareLieAlgebra.pdf ''Lie Algebra for the Poincaré, and Lorentz, Groups'']
*[http://arxiv.org/pdf/hep-th/0611263.pdf ''The unitary representations of the Poincaré group in any spacetime dimension'']
*{{cite web|title=McGraw-Hill dictionary of scientific and technical terms|url=http://www.answers.com/topic/irreducible-representation-of-a-group}}
 
[[Category:Group theory]]
[[Category:Representation theory]]
[[Category:Theoretical physics]]
[[Category:Theoretical chemistry]]
[[Category:Symmetry]]

Revision as of 19:06, 18 February 2014

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Aims people think a paid service won't disappear built in the path GR has, but that's not always fact. A paid service may well close shop or shutter it's doors , too. Personally, I'd rather pay get GR back.

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ESPN analyst Ron Jaworski, an old NFL quarterback and current partowner about the . How Tebow be slow and methodical in case your team never fully invested in him? Throughout his shortlived Nike Roshe Run Australia Cheap threeyear NFL career, Tebow have not possessed a chance .

05/14/2013 1:00 pm View LinkTim Tebow NY Jets RUINED His Career . Says NY Giants Owner Steve Tisch

The modern York Jets DESTROYED any chance Tim Tebow had to be a winning QB around the NFL . this reported by NY Giants coowner Steve Tisch. "I think him going inside Broncos about the Jets had been not in anybody's welfare," Tisch told TMZ on .

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know, I am badly for your guy, Tisch . Nike Shop Adelaide s probably a fine guy and not wii quarterback. So i wish him good luck. I sincerely wish him take care. Tebow has already just a few job offers while the Jets cut him. The general manager using the .

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The gridiron is a central theme to Tebow's roughly hourlong address, but he had not specify Air Max Shoes Australia anything about his current search for employment, nor would he accept questions inside media afterwards. When it was his decorated graduating high school or.

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The popular York Jets DESTROYED any chance Tim Tebow must be a triumphant QB throughout the NFL . this as indicated by NY Giants coowner Steve Tisch. "I think him .

05/13/2013 9:11 pm View WebsiteNY Giants coowner Steve Tisch feels detrimental Tim Tebow, says NY .

NY Giants coowner Steve Tisch feels detrimental to Tim Tebow, says NY Jets were bad fit for quarterback 'You know, I think badly just for the guy,' Tisch told TMZ.

Is Tim Tebow being blackballed with all the NFL? The fast and also reply is that Tim Tebow will not be suitable as a considerate player as being a host distraction inside of the NFL, and .

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The pastor could not be reached, but audio clips of several of his sermons are available on the church's website. On one, he blasts the late televangelist the Rev. Robert Schuller, saying his book, "Self Esteem: The brand new Reformation" is "a book that will one day be burned in the fires of Hell.".

Think Wikipedia and the separation of the various languages. I believe this analogy works great, and it might even be easy for you to implement, given the already broken up sister sites. The difference is, it is for espanol (pardon lack of tilde) speaking people.

Guys. Awaken. Why would we go to all this trouble just to get Firefox to operate properly? Why can't it simply work correctly right out of the box, like Chrome? Myself and I suspect many others are just about through trying to use Firefox in its currently unusable state.

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You will see a staggered start. Runners will start first and the walkers follows. Strollers and wheelchairs are welcome. I don think individuals need to get too tweaked about the end of the world coming. Quite frankly when it comes, if it is during any of our lifetimes, there's absolutely nothing we can do about it. Back in the 1830s a guy named William Miller started proclaiming the end of the world.

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