In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in the
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
s and
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
, an irreducible representation
or irrep of an algebraic structure
is a nonzero representation that has no proper nontrivial subrepresentation
, with
closed under the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of
.
Every finite-dimensional
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
on a
Hilbert space is the
direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular
unipotent
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''.
In particular, a square matrix ''M'' is a unipoten ...
matrices is indecomposable but reducible.
History
Group representation theory was generalized by
Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
from the 1940s to give
modular representation theory
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as ...
, in which the matrix operators act on a vector space over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of arbitrary
characteristic, rather than a vector space over the field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s or over the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. The structure analogous to an irreducible representation in the resulting theory is a
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cycl ...
.
Overview
Let
be a representation i.e. a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
of a group
where
is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. 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
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
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
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
subrepresentations (all representations can form a subrepresentation with the trivial
-invariant subspaces, e.g. the whole vector space
, and
). 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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, 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
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of matrices. As notation, let denote elements of a group with group product signified without any symbol, so is the group product of and and is also an element of , and let representations be indicated by . The representation of ''a'' is written
:
By definition of group representations, the representation of a group product is translated into
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
of the representations:
:
If is the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the group (so that , etc.), then is an
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, 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 is a
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
.
Reducible and irreducible representations
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, for example, of dimension 2, 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 representations
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:
:
which
diagonalizes every matrix in the representation into the same pattern of
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
block
Block or blocked may refer to:
Arts, entertainment and media Broadcasting
* Block programming, the result of a programming strategy in broadcasting
* W242BX, a radio station licensed to Greenville, South Carolina, United States known as ''96.3 ...
s. Each such block is then a group subrepresentation independent from the others. The representations and are said to be equivalent representations. The (''k''-dimensional, say) representation can be decomposed into a
direct sum of matrices:
:
so is decomposable, and it is customary to label the decomposed matrices by a superscript in brackets, as in for , although some authors just write the numerical label without parentheses.
The dimension of is the sum of the dimensions of the blocks:
:
If this is not possible, i.e. , then the representation is indecomposable.
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 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
is finite, and it has a representation over field
, then an indecomposable representation is an irreducible representation.
* 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 representations
Trivial representation
All groups
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
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about ...
. In particular, all complex representations decompose as a direct sum of irreps, and the number of irreps of
is equal to the number of conjugacy classes of
.
* The irreducible complex representations of
are exactly given by the maps
, where
is an
th
root of unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
.
* 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
.
* Let
be a group. The
regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation.
One distinguishes the left regular rep ...
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 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 chemistry
In
quantum physics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
and
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
, each set of
degenerate eigenstates of the
Hamiltonian operator
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
comprises a vector space 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
Split(s) or The Split may refer to:
Places
* Split, Croatia, the largest coastal city in Croatia
* Split Island, Canada, an island in the Hudson Bay
* Split Island, Falkland Islands
* Split Island, Fiji, better known as Hạfliua
Arts, enterta ...
under perturbations; or transition to other states in . 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 rule
In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in ...
s to be determined.
Lie groups
Lorentz group
The irreps of and , where is the generator of rotations and 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 equation
In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
s.
See also
Associative algebras
*
Simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cycl ...
*
Indecomposable module In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Jacobson (2009), p. 111.
Indecomposable is a weaker notion than simple module (which is also sometimes called irredu ...
*
Representation of an associative algebra
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint funct ...
Lie groups
*
Representation theory of Lie algebras
*
Representation theory of SU(2)
In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abel ...
*
Representation theory of SL2(R)
In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
Structure of the complexified Lie algebra
We choo ...
*
Representation theory of the Galilean group
In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin (normally explained in Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the ...
*
Representation theory of diffeomorphism groups
In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold ''M'' is the initial observation that (for ''M'' connected) that group acts transitively on ''M''.
History
A survey paper from 1975 of t ...
*
Representation theory of the Poincaré group
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics.
In a physical theor ...
*
Theorem of the highest weight In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra \mathfrak g. Theorems 9.4 and 9.5 There is a closely related theorem classifying the ...
References
Books
*
*
*
*
*
*
*
*
*
*
*
Articles
*
*
Further reading
*
External links
*
*
*
*
*
*
*
*, see chapter 40
*
*
*
*{{cite web, title=McGraw-Hill dictionary of scientific and technical terms, website=
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Group theory
Representation theory
Theoretical physics
Theoretical chemistry
Symmetry