Irreducible representation

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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 nontrivial subrepresentation , with closed under the action of .

Every finite-dimensional unitary representation on a Hilbert 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), the converse may not hold, such as the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices, which is indecomposable but reducible.

HistoryEdit

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.[citation needed]

OverviewEdit

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 co-restriction of   to the general linear group of 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 representationsEdit

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 statements correspond to the requirement that D is a group homomorphism.

Reducible and irreducible representationsEdit

A representation is reducible if it contains a nontrivial G-invariant subspace, that is to say, all the matrices   can be put in upper triangular block form by the same invertible matrix  . In other words, if there is a similarity transformation:

 

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 of dimension k, then we have:

 

where   is a nontrivial subrepresentation. If we are able to find a matrix   that makes   as well, then   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   above to the standard basis.

Decomposable and indecomposable representationsEdit

A representation is decomposable if all the matrices   can be put in block-diagonal form by the same invertible matrix  . In other words, if there is a similarity transformation:[1]

 

which diagonalizes every matrix in the representation into the same pattern of diagonal blocks. Each such block is then a group subrepresentation independent from the others. 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]

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   above to the standard basis.

Connection between irreducible representation and decomposable representationEdit

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   is finite, and it has a representation over field  , then an indecomposable representation is an irreducible representation. [4]
  • When group   is finite, and it has a representation over field  , if we have  , then an indecomposable representation is an irreducible representation.

Examples of irreducible representationsEdit

Trivial representationEdit

All groups   have a one-dimensional, irreducible trivial representation.

One-dimensional representationEdit

Any one-dimensional representation is irreducible by virtue since it has no proper nontrivial subspaces.

Irreducible complex representationsEdit

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  .[5]

  • 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 one-dimensional and isomorphic to the trivial representation of  . The latter is   dimensional and is known as the standard representation of  .[5]
  • 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  Edit

  • Let   be a   group and   be a finite dimensional irreducible representation of G over  . By Orbit-stabilizer theorem, the orbit of every   element acted by the   group   has size being power of  . Since the sizes of all these orbits sum up to the size of  , and   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   such that   for all  . This forces every irreducible representation of a   group over   to be one dimensional.

Applications in theoretical physics and chemistryEdit

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.[6]

Lie groupsEdit

Lorentz groupEdit

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.[7]

See alsoEdit

Associative algebrasEdit

Lie groupsEdit

ReferencesEdit

  1. ^ a b E. P. Wigner (1959). Group theory and its application to the quantum mechanics of atomic spectra. Pure and applied physics. Academic press. p. 73.
  2. ^ W. K. Tung (1985). Group Theory in Physics. World Scientific. p. 32. ISBN 978-997-1966-560.
  3. ^ W. K. Tung (1985). Group Theory in Physics. World Scientific. p. 33. ISBN 978-997-1966-560.
  4. ^ Artin, Michael (2nd ed). Algebra. Pearson. p. 295. ISBN 978-0132413770. Check date values in: |year= (help)
  5. ^ a b Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. Springer-Verlag. ISBN 978-0-387-90190-9.
  6. ^ "A Dictionary of Chemistry, Answers.com" (6th ed.). Oxford Dictionary of Chemistry.
  7. ^ 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:10.1016/0003-4916(92)90176-M.

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