“Irreducible representation”的意思、由来-开放百科全书

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

Every finite-dimensional unitary representation on a Hermitian{{clarify|pre-text=Sesquilinear form or|date=July 2016}} 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 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

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|date=July 2013}}

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 non-trivial 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 {{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

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

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

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

Decomposable and Indecomposable representations

A representation is decomposable if a similar matrix {{math|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 representation of the group independent of each other. The representations {{math|D(a)}} and {{math|D′(a)}} are said to be equivalent representations.^[2] The representation can be decomposed into a direct sum of k matrices:

so {{math|D(a)}} is decomposable, and it is customary to label the decomposed matrices by a superscript in brackets, as in {{math|D⁽ⁿ⁾(a)}} for {{math|n {{=}} 1, 2, ..., k}}, although some authors just write the numerical label without parentheses.

If this is not possible, i.e. {{math|k{{=}}1}}, then the representation is indecomposable.^[1]^[3]

Examples of Irreducible Representations

Trivial Representation

All groups

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

is equal to the number of conjugacy classes of

.^[4]

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 {{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 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 selection rules to be determined.^[5]

Lie groups

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.^[6]

See also

Associative algebras

Lie groups

References

1. ^¹{{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=}}
2. ^{{cite book|author= W.K. Tung|title=Group Theory in Physics|page=32|publisher=World Scientific|year=1985|url=https://books.google.com/?id=O89tgpOBO04C&printsec=frontcover&dq=group+theory+in+physics#v=onepage&q=group%20theory%20in%20physics&f=false|isbn=978-997-1966-560}}
3. ^{{cite book |author= W.K. Tung|title=Group Theory in Physics |page=33|publisher=World Scientific|year=1985|url=https://books.google.com/?id=O89tgpOBO04C&printsec=frontcover&dq=group+theory+in+physics#v=onepage&q=group%20theory%20in%20physics&f=false|isbn=978-997-1966-560}}
4. ^¹{{cite book| authorlink=Jean-Pierre Serre| first=Jean-Pierre| last= Serre| title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | isbn=978-0387901909}}
5. ^{{cite web|publisher=Oxford Dictionary of Chemistry|title=A Dictionary of Chemistry, Answers.com| edition=6th |url= http://www.answers.com/topic/irreducible-representation}}
6. ^{{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 | url = http://www.sciencedirect.com/science/article/pii/000349169290176M | volume=216 |issue=2 | pages=226–267| bibcode=1992AnPhy.216..226J}}

History

Overview

Notation and terminology of group representations

Decomposable and Indecomposable representations

Examples of Irreducible Representations

Trivial Representation

Irreducible Complex Representations

Applications in theoretical physics and chemistry

Lie groups

Lorentz group

See also

Associative algebras

Lie groups

References

Books

Papers

Further reading

External links