Irreducible representation

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In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation $(\rho ,V)$ or irrep of an algebraic structure $A$ is a nonzero representation that has no proper subrepresentation $(\rho |_{W},W),W\subset V$ closed under the action of $\{\rho (a):a\in A\}$.

Every finite-dimensional unitary representation on a Hermitian vector space $V$ 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.

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 $K$ 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

Further information: Group representation

Let $\rho$ be a representation i.e. a homomorphism $\rho :G\to GL(V)$ of a group $G$ where $V$ is a vector space over a field $F$. If we pick a basis $B$ for $V$, $\rho$ 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 $V$ without a basis.

A linear subspace $W\subset V$ is called $G$-invariant if $\rho (g)w\in W$ for all $g\in G$ and all $w\in W$. The restriction of $\rho$ to a $G$-invariant subspace $W\subset V$ is known as a subrepresentation. A representation $\rho :G\to GL(V)$ is said to be irreducible if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial $G$-invariant subspaces, e.g. the whole vector space $V$, and {0}). If there is a proper non-trivial invariant subspace, $\rho$ 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

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

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

$D(ab)=D(a)D(b)$

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

$D(ea)=D(ae)=D(a)D(e)=D(e)D(a)=D(a)$

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:

$D'(a)\equiv P^{-1}D(a)P$

which diagonalizes every matrix in the representation into the same pattern of diagonal blocks – each of the blocks are representation of the group independent of each other. The representations D(a) and D'(a) are said to be equivalent representations. The representation can be decomposed into a direct sum of k matrices:

$D'(a)=P^{-1}D(a)P={\begin{pmatrix}D^{(1)}(a)&0&\cdots &0\\0&D^{(2)}(a)&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &D^{(k)}(a)\\\end{pmatrix}}=D^{(1)}(a)\oplus D^{(2)}(a)\oplus \cdots \oplus D^{(k)}(a)$

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:

$\mathrm {dim} [D(a)]=\mathrm {dim} [D^{(1)}(a)]+\mathrm {dim} [D^{(2)}(a)]+\ldots +\mathrm {dim} [D^{(k)}(a)]$

If this is not possible, i.e. k=1, then the representation is indecomposable.

Examples of Irreducible Representations

Trivial Representation

All groups $G$ have a one-dimensional, irreducible trivial representation. More generally, any one-dimensional 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 $G$ is equal to the number of conjugacy classes of $G$.

• The irreducible complex representations of $\mathbb {Z} /n\mathbb {Z}$ are exactly given by the maps $1\mapsto \gamma$, where $\gamma$ is an $n$th root of unity.
• Let $V$ be an $n$-dimensional complex representation of $S_{n}$ with basis $\{v_{i}\}_{i=1}^{n}$. Then $V$ decomposes as a direct sum of the irreps $V_{triv}=\mathbb {C} (\sum _{i=1}^{n}v_{i})$ and the orthogonal subspace given by:

$V_{std}=\{\sum _{i=1}^{n}a_{i}v_{i}:a_{i}\in \mathbb {C} ,\sum _{i=1}^{n}a_{i}=0\}$

The former irrep is one-dimensional and isomorphic to the trivial representation of $S_{n}$. The latter is $n-1$ dimensional and is known as the standard representation of $S_{n}$.

• Let $G$ be a group. The regular representation of $G$ is the free complex vector space on the basis $\{e_{g}\}_{g\in G}$ with the group action $g\cdot e_{g'}=e_{gg'}$, denoted $\mathbb {C} G$. All irreducible representations of $G$ appear in the decomposition of $\mathbb {C} G$ as a direct sum of irreps.

Applications in theoretical physics and chemistry

See also: Symmetry in quantum mechanics and Jahn–Teller effect

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.

Lie groups

Main article: Representation theory of Lie groups

Lorentz group

Main article: Representation theory of the 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.

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)
• Representation theory of the Galilean group
• Representation theory of diffeomorphism groups
• Representation theory of the Poincaré group
• Theorem of the highest weight

References

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 978-0-07-084011-9.
• V. Heine (2007). Group theory in quantum mechanics: an introduction to its present usage. Dover. ISBN 978-0-07-084011-9.
• V. Heine (1993). Group Theory in Quantum Mechanics: An Introduction to Its Present Usage. Courier Dover Publications. ISBN 978-048-6675-855.

• E. Abers (2004). Quantum Mechanics. Addison Wesley. p. 425. ISBN 978-0-13-146100-0.
• B. R. Martin, G.Shaw. Particle Physics (3rd ed.). Manchester Physics Series, John Wiley & Sons. p. 3. ISBN 978-0-470-03294-7.
• Weinberg, S (1995), The Quantum Theory of Fields, 1, Cambridge university press, pp. 230–231, ISBN 978-0-521-55001-7
• Weinberg, S (1996), The Quantum Theory of Fields, 2, Cambridge university press, ISBN 978-0-521-55002-4
• Weinberg, S (2000), The Quantum Theory of Fields, 3, Cambridge university press, ISBN 978-0-521-66000-6
• R. Penrose (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4.
• P. W. Atkins (1970). Molecular Quantum Mechanics (Parts 1 and 2): An introduction to quantum chemistry. 1. Oxford University Press. pp. 125–126. ISBN 978-0-19-855129-4.

Papers

• 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. 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 2012-06-17.
• Bekaert, Xavier; Boulanger, Niclas (2006). "The unitary representations of the Poincaré group in any spacetime dimension". arXiv:hep-th/0611263.
• "McGraw-Hill dictionary of scientific and technical terms".