Irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper subrepresentation closed under the action of .
Algebraic structure → Group theory Group theory 



Infinite dimensional Lie group

Every finitedimensional unitary representation on a Hermitian vector space 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 twodimensional representation of the real numbers acting by upper triangular unipotent matrices.
History
Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field of arbitrary characteristic, rather than a vector space over the field of real numbers or over the field of complex numbers. The structure analogous to an irreducible representation in the resulting theory is a simple module.
Overview
Let be a representation i.e. a homomorphism of a group where is a vector space over a field . If we pick a basis for , 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 without a basis.
A linear subspace is called invariant if for all and all . The restriction of to a invariant subspace is known as a subrepresentation. A representation is said to be irreducible if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial invariant subspaces, e.g. the whole vector space , and {0}). If there is a proper nontrivial invariant subspace, is said to be reducible.
Notation and terminology of group representations
Group elements can be represented by 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 a, b, c... denote elements of a group G with group product signified without any symbol, so ab is the group product of a and b and is also an element of G, and let representations be indicated by D. The representation of a is written
By definition of group representations, the representation of a group product is translated into matrix multiplication of the representations:
If e is the identity element of the group (so that ae = ea = a, etc.), then D(e) is an identity matrix, or identically a block matrix of identity matrices, since we must have
and similarly for all other group elements. The last two staments correspond to the requirement that D is a group homomorphism.
Decomposable and indecomposable representations
A representation is decomposable if a similar matrix P can be found for the similarity transformation:[1]
which diagonalizes every matrix in the representation into the same pattern of diagonal blocks – each of the blocks are representations of the group independent of each other. The representations D(a) and D′(a) are said to be equivalent representations.[2] The representation can be decomposed into a direct sum of k > 1 matrices:
so D(a) is decomposable, and it is customary to label the decomposed matrices by a superscript in brackets, as in D^{(n)}(a) for n = 1, 2, ..., k, although some authors just write the numerical label without parentheses.
The dimension of D(a) is the sum of the dimensions of the blocks:
If this is not possible, i.e. k = 1, then the representation is indecomposable.[1][3]
Examples of irreducible representations
Trivial representation
All groups have a onedimensional, irreducible trivial representation. More generally, any onedimensional representation is irreducible by virtue of having no proper nontrivial subspaces.
Irreducible complex representations
The irreducible complex representations of a finite group G can be characterized using results from character theory. In particular, all such representations decompose as a direct sum of irreps, and the number of irreps of is equal to the number of conjugacy classes of .[4]
 The irreducible complex representations of are exactly given by the maps , where is an th root of unity.
 Let be an dimensional complex representation of with basis . Then decomposes as a direct sum of the irreps
 and the orthogonal subspace given by
 The former irrep is onedimensional and isomorphic to the trivial representation of . The latter is dimensional and is known as the standard representation of .[4]
 Let be a group. The regular representation of is the free complex vector space on the basis with the group action , denoted All irreducible representations of appear in the decomposition of as a direct sum of irreps.
Example of an irreducible representation over
 Let be a group and be a finite dimensional irreducible representation of G over . By the theory of group actions, the set of fixed points of is non empty, that is, there exists some such that for all . This forces every irreducible representation of a group over to be one dimensional.
Applications in theoretical physics and chemistry
In quantum physics and quantum chemistry, each set of degenerate eigenstates of the Hamiltonian operator comprises a vector space 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 split under perturbations; or transition to other states in 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 selection rules to be determined.[5]
Lie groups
Lorentz group
The irreps of D(K) and D(J), where J is the generator of rotations and 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.[6]
See also
Associative algebras
References
 E. P. Wigner (1959). Group theory and its application to the quantum mechanics of atomic spectra. Pure and applied physics. Academic press. p. 73.
 W. K. Tung (1985). Group Theory in Physics. World Scientific. p. 32. ISBN 9789971966560.
 W. K. Tung (1985). Group Theory in Physics. World Scientific. p. 33. ISBN 9789971966560.
 Serre, JeanPierre (1977). Linear Representations of Finite Groups. SpringerVerlag. ISBN 9780387901909.
 "A Dictionary of Chemistry, Answers.com" (6th ed.). Oxford Dictionary of Chemistry.
 T. Jaroszewicz; P. S. Kurzepa (1992). "Geometry of spacetime propagation of spinning particles". Annals of Physics. 216 (2): 226–267. Bibcode:1992AnPhy.216..226J. doi:10.1016/00034916(92)90176M.
Books
 H. Weyl (1950). The theory of groups and quantum mechanics. Courier Dover Publications. p. 203. ISBN 9780486602691.
 A. D. Boardman; D. E. O'Conner; P. A. Young (1973). Symmetry and its applications in science. McGraw Hill. ISBN 9780070840119.
 V. Heine (2007). Group theory in quantum mechanics: an introduction to its present usage. Dover. ISBN 9780070840119.
 V. Heine (1993). Group Theory in Quantum Mechanics: An Introduction to Its Present Usage. Courier Dover Publications. ISBN 9780486675855.
 E. Abers (2004). Quantum Mechanics. Addison Wesley. p. 425. ISBN 9780131461000.
 B. R. Martin, G.Shaw. Particle Physics (3rd ed.). Manchester Physics Series, John Wiley & Sons. p. 3. ISBN 9780470032947.
 Weinberg, S. (1995), The Quantum Theory of Fields, 1, Cambridge university press, pp. 230–231, ISBN 9780521550017
 Weinberg, S. (1996), The Quantum Theory of Fields, 2, Cambridge university press, ISBN 9780521550024
 Weinberg, S. (2000), The Quantum Theory of Fields, 3, Cambridge university press, ISBN 9780521660006
 R. Penrose (2007). The Road to Reality. Vintage books. ISBN 9780679776314.
 P. W. Atkins (1970). Molecular Quantum Mechanics (Parts 1 and 2): An introduction to quantum chemistry. 1. Oxford University Press. pp. 125–126. ISBN 9780198551294.
Articles
 Bargmann, V.; Wigner, E. P. (1948). "Group theoretical discussion of relativistic wave equations". Proc. Natl. Acad. Sci. U.S.A. 34 (5): 211–23. Bibcode:1948PNAS...34..211B. doi:10.1073/pnas.34.5.211. PMC 1079095. PMID 16578292.
 E. Wigner (1937). "On Unitary Representations Of The Inhomogeneous Lorentz Group" (PDF). Annals of Mathematics. 40 (1): 149–204. Bibcode:1989NuPhS...6....9W. doi:10.2307/1968551. JSTOR 1968551.
Further reading
 Artin, Michael (1999). "Noncommutative Rings" (PDF). Chapter V.
External links
 "Commission on Mathematical and Theoretical Crystallography, Summer Schools on Mathematical Crystallography" (PDF). 2010.
 van Beveren, Eef (2012). "Some notes on group theory" (PDF).
 Teleman, Constantin (2005). "Representation Theory" (PDF).
 Finley. "Some Notes on Young Tableaux as useful for irreps of su(n)" (PDF).
 Hunt (2008). "Irreducible Representation (IR) Symmetry Labels" (PDF).
 Dermisek, Radovan (2008). "Representations of Lorentz Group" (PDF).
 Maciejko, Joseph (2007). "Representations of Lorentz and Poincaré groups" (PDF).
 Woit, Peter (2015). "Quantum Mechanics for Mathematicians: Representations of the Lorentz Group" (PDF)., see chapter 40
 Drake, Kyle; Feinberg, Michael; Guild, David; Turetsky, Emma (2009). "Representations of the Symmetry Group of Spacetime" (PDF).
 Finley. "Lie Algebra for the Poincaré, and Lorentz, Groups" (PDF). Archived from the original (PDF) on 20120617.
 Bekaert, Xavier; Boulanger, Niclas (2006). "The unitary representations of the Poincaré group in any spacetime dimension". arXiv:hepth/0611263.
 "McGrawHill dictionary of scientific and technical terms".