View source for Irreducible representation

{{Short description|Type of group and algebra representation}}

In [[Mathematics|mathematics]], specifically in the [[Representation theory|representation theory]] of [[Group (mathematics)|group]]s and [[Algebra over a field|algebras]], an '''irreducible representation''' <math>(\rho, V)</math> or '''irrep''' of an algebraic structure <math>A</math> is a nonzero representation that has no proper nontrivial subrepresentation <math>(\rho|_W,W)</math>, with <math>W \subset V</math> closed under the [[Group action|action]] of <math>\{ \rho(a) : a\in A \}</math>.

Every finite-dimensional [[Unitary representation|unitary representation]] on a [[Hilbert space]] <math>V</math> is the [[Direct sum|direct sum]] of irreducible representations. Irreducible representations are always '''indecomposable''' (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular [[Unipotent|unipotent]] matrices is indecomposable but reducible.

==History==

Group representation theory was generalized by [[Biography:Richard Brauer|Richard Brauer]] from the 1940s to give [[Modular representation theory|modular representation theory]], in which the matrix operators act on a vector space over a [[Field (mathematics)|field]] <math>K</math> of arbitrary [[Characteristic (algebra)|characteristic]], rather than a vector space over the field of [[Real number|real number]]s or over the field of [[Complex number|complex number]]s. The structure analogous to an irreducible representation in the resulting theory is a [[Simple module|simple module]].{{citation needed|date=July 2013}}

==Overview==

{{details|Group representation}}

Let <math>\rho</math> be a representation i.e. a [[Homomorphism|homomorphism]] <math>\rho: G \to GL(V)</math> of a group <math>G</math> where <math>V</math> is a [[Vector space|vector space]] over a [[Field (mathematics)|field]] <math>F</math>. If we pick a basis <math>B</math> for <math>V</math>, <math>\rho</math> can be thought of as a function (a homomorphism) from a group into 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</math> without a basis.

A [[Linear subspace|linear subspace]] <math>W\subset V</math> is called '''<math>G</math>-invariant''' if <math>\rho(g)w\in W</math> for all <math>g\in G</math> and all <math> w\in W</math>. The co-restriction of <math>\rho</math> to the general linear group of a <math>G</math>-invariant subspace <math>W\subset V</math> is known as a '''subrepresentation'''. A representation <math>\rho: G \to GL(V)</math> is said to be '''irreducible''' if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial <math>G</math>-invariant subspaces, e.g. the whole vector space <math>V</math>, and {0}). If there is a proper nontrivial invariant subspace, <math>\rho</math> is said to be '''reducible'''.

===Notation and terminology of group representations===

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|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 as

:<math>D(a) = \begin{pmatrix}
D(a)_{11} & D(a)_{12} & \cdots & D(a)_{1n} \\
D(a)_{21} & D(a)_{22} & \cdots & D(a)_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
D(a)_{n1} & D(a)_{n2} & \cdots & D(a)_{nn} \\
\end{pmatrix}</math>

By definition of group representations, the representation of a group product is translated into [[Matrix multiplication|matrix multiplication]] of the representations:

:<math>D(ab) = D(a)D(b) </math>

If {{math|''e''}} is the [[Identity element|identity element]] of the group (so that {{math|1=''ae'' = ''ea'' = ''a''}}, etc.), then {{math|''D''(''e'')}} is an [[Identity matrix|identity matrix]], or identically a block matrix of identity matrices, since we must have

:<math>D(ea) = D(ae) = D(a)D(e) = D(e)D(a) = D(a)</math>

and similarly for all other group elements. The last two statements correspond to the requirement that {{math|''D''}} is a [[Group homomorphism|group homomorphism]].

=== Reducible and irreducible representations ===
A representation is reducible if it contains a nontrivial G-invariant subspace, that is to say, all the matrices <math>D(a)</math> can be put in upper triangular block form by the same invertible matrix <math>P</math>. In other words, if there is a similarity transformation:
:<math> D'(a) \equiv P^{-1} D(a) P,</math>
which maps every matrix in the representation into the same pattern upper triangular blocks. Every ordered sequence minor block is a group subrepresentation. That is to say, if the representation is, for example, of dimension 2, then we have:
<math display="block">D'(a) = P^{-1} D(a) P = \begin{pmatrix} 
D^{(11)}(a) & D^{(12)}(a)  \\
0 & D^{(22)}(a)
\end{pmatrix}, </math>

where <math>D^{(11)}(a)</math> is a nontrivial subrepresentation. If we are able to find a matrix <math>P </math> that makes <math>D^{(12)}(a) = 0</math> as well, then <math>D(a)</math> is not only reducible but also decomposable.

'''Notice:''' Even if a representation is reducible, its matrix representation may still not be the upper triangular block form.  It will only have this form if we choose a suitable basis, which can be obtained by applying the matrix <math>P^{-1}</math> above to the standard basis.

===Decomposable and indecomposable representations===

A representation is decomposable if all the matrices <math>D(a)</math> can be put in block-diagonal form by the same invertible matrix <math>P</math>. In other words, if there is a [[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 }}</ref>

:<math> D'(a) \equiv P^{-1} D(a) P,</math>

which diagonalizes every matrix in the representation into the same pattern of [[Diagonal matrix|diagonal]] [[Block matrix|block]]s.&nbsp;Each such block is then a group subrepresentation independent from the others. The representations {{math|''D''(''a'')}} and {{math|''D′''(''a'')}} 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=https://books.google.com/books?id=O89tgpOBO04C&q=group+theory+in+physics |isbn=978-997-1966-560}}</ref> The (''k''-dimensional, say) representation can be decomposed into a direct sum of {{math|''k'' > 1}} matrices:

:<math>D'(a) = P^{-1} D(a) P = \begin{pmatrix} 
D^{(1)}(a) & 0 & \cdots & 0 \\
0 & D^{(2)}(a) & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & D^{(k)}(a) \\
\end{pmatrix} = D^{(1)}(a) \oplus D^{(2)}(a) \oplus \cdots \oplus D^{(k)}(a),</math>

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|1=''n'' = 1, 2, ..., ''k''}}, although some authors just write the numerical label without parentheses.

The dimension of {{math|''D''(''a'')}} is the sum of the dimensions of the blocks:

:<math>\dim[D(a)] = \dim[D^{(1)}(a)] + \dim[D^{(2)}(a)] + \cdots + \dim[D^{(k)}(a)].</math>

If this is not possible, i.e. {{math|1=''k'' = 1}}, 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 |page=33 |publisher=World Scientific |year=1985 |url=https://books.google.com/books?id=O89tgpOBO04C&q=group+theory+in+physics |isbn=978-997-1966-560}}</ref>

'''Notice''': Even if a representation is decomposable, its matrix representation may not be the diagonal block form.  It will only have this form if we choose a suitable basis, which can be obtained by applying the matrix <math>P^{-1}</math> above to the standard basis.

=== Connection between irreducible representation and indecomposable representation ===
An irreducible representation is by nature an indecomposable one. However, the converse may fail. 

But under some conditions, we do have an indecomposable representation being an irreducible representation. 

* When group <math>G</math> is finite, and it has a representation over field <math>\Complex</math>, then an indecomposable representation is an irreducible representation. <ref>{{Cite book|last=Artin|first=Michael|title=Algebra|publisher=Pearson| year=2011| edition=2nd |isbn=978-0132413770|pages=295}}</ref>
* When group <math>G</math> is finite, and it has a representation over field <math>K</math>, if we have <math>char(K)\nmid |G|</math>, then an indecomposable representation is an irreducible representation.

==Examples of irreducible representations==

===Trivial representation===
All groups <math>G</math> have a one-dimensional, irreducible trivial representation by mapping all group elements to the identity transformation.

=== One-dimensional representation ===
Any one-dimensional representation is irreducible since it has no proper nontrivial subspaces.

===Irreducible complex representations===

The irreducible complex representations of a finite group G can be characterized using results from [[Character theory|character theory]]. In particular, all complex representations decompose as a direct sum of irreps, and the number of irreps of <math>G</math> is equal to the number of conjugacy classes of <math>G</math>.<ref name="Serre">{{cite book|  first=Jean-Pierre| last= Serre| title=Linear Representations of Finite Groups | url=https://archive.org/details/linearrepresenta1977serr| url-access=registration| publisher=Springer-Verlag | year=1977 | isbn=978-0-387-90190-9}}</ref>

* The irreducible complex representations of <math>\Z / n\Z</math> are exactly given by the maps <math>1 \mapsto \gamma</math>, where <math>\gamma</math> is an <math>n</math>th [[Root of unity|root of unity]].
* Let <math>V</math> be an <math>n</math>-dimensional complex representation of <math>S_n</math> with basis <math>\{v_i\}^n_{i=1}</math>. Then <math>V</math> decomposes as a direct sum of the irreps <math display="block">V_\text{triv} = \Complex \left ( \sum^n_{i=1} v_i \right )</math> and the orthogonal subspace given by <math display="block">V_\text{std} = \left \{ \sum^n_{i=1} a_i v_i : a_i \in \Complex, \sum^n_{i=1} a_i = 0 \right \}.</math> The former irrep is one-dimensional and isomorphic to the trivial representation of <math>S_n</math>. The latter is <math>n-1</math> dimensional and is known as the standard representation of <math>S_n</math>.<ref name="Serre"/>
* Let <math>G</math> be a group. The [[Regular representation|regular representation]] of <math>G</math> is the free complex vector space on the basis <math>\{e_g\}_{g \in G}</math> with the group action <math>g \cdot e_{g'} = e_{gg'}</math>, denoted <math>\Complex G.</math> All irreducible representations of <math>G</math> appear in the decomposition of <math>\Complex G</math> as a direct sum of irreps.

===Example of an irreducible representation over {{math|'''F'''<sub>''p''</sub>}}===
*Let <math>G</math> be a <math>p</math> group and <math>V = \mathbb{F}_p^{n}</math> be a finite dimensional irreducible representation of G over <math>\mathbb{F}_p</math>. By Orbit-stabilizer theorem, the orbit of every <math>V</math> element acted by the <math>p</math> group <math>G</math> has size being power of <math>p</math>. Since the sizes of all these orbits sum up to the size of <math>G</math>, and  <math>0 \in V</math> is in a size 1 orbit only containing itself, there must be other orbits of size 1 for the sum to match. That is, there exists some <math>v\in V</math> such that <math>gv = v</math> for all <math>g \in G</math>. This forces every irreducible representation of a  <math>p</math> group over <math> \mathbb{F}_p</math> to be one dimensional.

==Applications in theoretical physics and chemistry==


In quantum physics and [[Chemistry:Quantum chemistry|quantum chemistry]], each set of [[Physics:Degenerate energy levels|degenerate eigenstates]] of the [[Physics:Hamiltonian operator|Hamiltonian operator]] comprises a vector space {{mvar|V}} for a representation of the symmetry group of the Hamiltonian, a "multiplet", best studied through reduction to its irreducible parts. Identifying the irreducible representations therefore allows one to label the states, predict how they will [[Physics:Energy level splitting|split]] under perturbations; or transition to other states in {{mvar|V}}. Thus, in quantum mechanics, irreducible representations of the symmetry group of the system partially or completely label the energy levels of the system, allowing the [[Physics:Selection rule|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>{{better source needed|date=August 2022}}

== Lie groups ==


===Lorentz group===
{{main|Physics:Representation theory of the Lorentz group}}

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 equations.<ref>{{cite journal |author1=T. Jaroszewicz |author2=P. S. Kurzepa |year = 1992 |title = Geometry of spacetime propagation of spinning particles |journal = Annals of Physics |doi = 10.1016/0003-4916(92)90176-M |volume=216 |issue=2 |pages=226–267 |bibcode=1992AnPhy.216..226J}}</ref>

==See also==

===Associative algebras===

*[[Simple module]]
*[[Indecomposable module]]
*Representation of an associative algebra

===Lie groups===

* Representation theory of Lie algebras
* [[Representation theory of SU(2)]] 
* [[Representation theory of SL2(R)]] 
* [[Physics:Representation theory of the Galilean group|Representation theory of the Galilean group]]
* [[Representation theory of diffeomorphism groups]]
* [[Representation theory of the Poincaré group]]
* [[Theorem of the highest weight]]

==References==

{{reflist}}

===Books===

* {{cite book|author=H. Weyl|title=The theory of groups and quantum mechanics|page=[https://archive.org/details/theoryofgroupsqu1950weyl/page/203 203]|publisher=Courier Dover Publications| year=1950 | url=https://archive.org/details/theoryofgroupsqu1950weyl |url-access=registration|quote=magnetic moments in relativistic quantum mechanics.|isbn=978-0-486-60269-1|}}
* {{cite book|author1=P. R. Bunker |author2=Per Jensen |title=Fundamentals of molecular symmetry|publisher=CRC Press|year=2004 |isbn=0-7503-0941-5}}[https://www.routledge.com/Fundamentals-of-Molecular-Symmetry/Bunker-Jensen/p/book/9780750309417]
* {{cite book|author1=A. D. Boardman |author2=D. E. O'Conner |author3=P. A. Young |title=Symmetry and its applications in science|publisher=McGraw Hill|year=1973|isbn=978-0-07-084011-9}}
* {{cite book|author=V. Heine|title=Group theory in quantum mechanics: an introduction to its present usage|publisher=Dover|year=2007 |url= https://archive.org/details/GroupTheoryInQuantumMechanics|isbn=978-0-07-084011-9}}
* {{cite book|author=V. Heine|title=Group Theory in Quantum Mechanics: An Introduction to Its Present Usage| publisher= Courier Dover Publications |year=1993 |url= https://books.google.com/books?id=NayFD34uEu0C&q=lorentz+group+in+relativistic+quantum+mechanics&pg=PA363|isbn=978-048-6675-855}}

* {{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|date=3 December 2008|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 = [https://archive.org/details/quantumtheoryoff00stev/page/230 230–231]|volume = 1|publisher = Cambridge university press|isbn = 978-0-521-55001-7|url-access = registration|url = https://archive.org/details/quantumtheoryoff00stev/page/230}}
* {{citation|last = Weinberg|first = S.|year = 1996|title = The Quantum Theory of Fields|volume = 2|publisher = Cambridge university press|isbn = 978-0-521-55002-4|url-access = registration|url = https://archive.org/details/quantumtheoryoff00stev}}
* {{citation|last = Weinberg|first = S.|year = 2000|title = The Quantum Theory of Fields|volume = 3|publisher = Cambridge university press|isbn = 978-0-521-66000-6|url-access = registration|url = https://archive.org/details/quantumtheoryoff00stev}}
* {{cite book|author=R. Penrose| title=The Road to Reality| publisher= Vintage books| year=2007 | isbn=978-0-679-77631-4| title-link=The Road to Reality}}
* {{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=978-0-19-855129-4}}

===Articles===

* {{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|issue=5|bibcode = 1948PNAS...34..211B |doi = 10.1073/pnas.34.5.211|pmid=16578292|pmc=1079095|doi-access=free}}
* {{cite journal| author = E. Wigner| year = 1937| title = On Unitary Representations Of The Inhomogeneous Lorentz Group| journal = Annals of Mathematics| volume = 40| pages = 149–204| number = 1| doi = 10.2307/1968551| bibcode = 1939AnMat..40..149W| jstor = 1968551| mr = 1503456| s2cid = 121773411| url = http://courses.theophys.kth.se/SI2390/wigner_1939.pdf| access-date = 2013-07-07| archive-date = 2015-10-04| archive-url = https://web.archive.org/web/20151004025027/http://courses.theophys.kth.se/SI2390/wigner_1939.pdf| url-status = dead}}

==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==
*{{cite web|url=http://www.crystallography.fr/mathcryst/pdf/nancy2010/Aroyo_reps2010.pdf|year=2010 | title=Commission on Mathematical and Theoretical Crystallography, Summer Schools on Mathematical Crystallography}}
*{{cite web|first1=Eef|last1=van Beveren|year=2012|url=http://cft.fis.uc.pt/eef/evbgroups.pdf|title=Some notes on group theory|access-date=2013-07-07|archive-url=https://web.archive.org/web/20110520062419/http://cft.fis.uc.pt/eef/evbgroups.pdf|archive-date=2011-05-20|url-status=dead}}
*{{cite web|url=http://math.berkeley.edu/~teleman/math/RepThry.pdf | first1=Constantin |last1=Teleman | title=Representation Theory | year =2005}}
*{{cite web|url=http://www.huntresearchgroup.org.uk/teaching/teaching_MOs_year2/L2_Addn_Symm_Labels.pdf|title=Irreducible Representation (IR) Symmetry Labels |last1=Hunt |year=2008}}
*{{cite web| url=http://www.physics.indiana.edu/~dermisek/QFT_08/qft-I-14-4p.pdf| title=Representations of Lorentz Group| year=2008| first1=Radovan| last1=Dermisek| access-date=2013-07-07| archive-url=https://web.archive.org/web/20181123185704/http://www.physics.indiana.edu/~dermisek/QFT_08/qft-I-14-4p.pdf| archive-date=2018-11-23| url-status=dead}}
*{{cite web|url=http://einrichtungen.ph.tum.de/T30f/lec/QFT/groups.pdf|title=Representations of Lorentz and Poincaré groups|first1=Joseph |last1= Maciejko |year=2007}}
*{{cite web|url=http://www.math.columbia.edu/~woit/QM/qmbook.pdf|first1=Peter |last1=Woit| year=2015|title=Quantum Mechanics for Mathematicians: Representations of the Lorentz Group}}, see chapter 40
*{{cite web|url=http://pages.cs.wisc.edu/~guild/symmetrycompsproject.pdf |title=Representations of the Symmetry Group of Spacetime | year=2009 |first1=Kyle | last1=Drake|first2=Michael |last2=Feinberg|first3=David|last3=Guild|first4=Emma | last4=Turetsky}}
*{{cite web|url= http://panda.unm.edu/Courses/Finley/P495/handouts/PoincareLieAlgebra.pdf |title = Lie Algebra for the Poincaré, and Lorentz, Groups |last1 = Finley |url-status = dead |archive-url = https://web.archive.org/web/20120617020207/http://panda.unm.edu/Courses/Finley/P495/handouts/PoincareLieAlgebra.pdf |archive-date = 2012-06-17 }}
*{{cite arXiv|eprint=hep-th/0611263|title=The unitary representations of the Poincaré group in any spacetime dimension |year=2006|first1=Xavier |last1=Bekaert|first2=Niclas|last2=Boulanger}}
*{{cite web|title=McGraw-Hill dictionary of scientific and technical terms|website=Answers.com |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]]

{{Sourceattribution|Irreducible representation}}