In the
mathematical field of
representation theory, group representations describe abstract
groups
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 ...
in terms of
bijective linear transformations of a
vector space to itself (i.e. vector space
automorphisms); in particular, they can be used to represent group elements as
invertible matrices so that the group operation can be represented by
matrix multiplication.
In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.
Representations of groups are important because they allow many
group-theoretic problems to be reduced to problems in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, which is well understood. They are also important in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
because, for example, they describe how the
symmetry group of a physical system affects the solutions of equations describing that system.
The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a
homomorphism from the group to the
automorphism group of an object. If the object is a vector space we have a ''linear representation''. Some people use ''realization'' for the general notion and reserve the term ''representation'' for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.
Branches of group representation theory
The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are:
*''
Finite groups'' — Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to
crystallography and to geometry. If the
field of scalars of the vector space has
characteristic ''p'', and if ''p'' divides the order of the group, then this is called ''
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 ha ...
''; this special case has very different properties. See
Representation theory of finite groups.
*''
Compact groups or
locally compact groups'' — Many of the results of finite group representation theory are proved by averaging over the group. These proofs can be carried over to infinite groups by replacement of the average with an integral, provided that an acceptable notion of integral can be defined. This can be done for locally compact groups, using
Haar measure. The resulting theory is a central part of
harmonic analysis. The
Pontryagin duality describes the theory for commutative groups, as a generalised
Fourier transform. See also:
Peter–Weyl theorem.
*''
Lie groups'' — Many important Lie groups are compact, so the results of compact representation theory apply to them. Other techniques specific to Lie groups are used as well. Most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields. See
Representations of Lie groups and
Representations of Lie algebras.
*''
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I ...
s'' (or more generally ''affine
group schemes'') — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different (and much less well understood). The analytic techniques used for studying Lie groups must be replaced by techniques from
algebraic geometry, where the relatively weak
Zariski topology causes many technical complications.
*''Non-compact topological groups'' — The class of non-compact groups is too broad to construct any general representation theory, but specific special cases have been studied, sometimes using ad hoc techniques. The ''semisimple Lie groups'' have a deep theory, building on the compact case. The complementary ''solvable'' Lie groups cannot be classified in the same way. The general theory for Lie groups deals with
semidirect products of the two types, by means of general results called ''
Mackey theory'', which is a generalization of
Wigner's classification methods.
Representation theory also depends heavily on the type of
vector space on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a
Hilbert space,
Banach space, etc.).
One must also consider the type of
field over which the vector space is defined. The most important case is the field of
complex numbers. The other important cases are the field of
real numbers,
finite fields, and fields of
p-adic numbers. In general,
algebraically closed fields are easier to handle than non-algebraically closed ones. The
characteristic of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the
order of the group.
Definitions
A representation of a
group ''G'' on a
vector space ''V'' over a
field ''K'' is a
group homomorphism from ''G'' to GL(''V''), the
general linear group on ''V''. That is, a representation is a map
:
such that
:
Here ''V'' is called the representation space and the dimension of ''V'' is called the dimension of the representation. It is common practice to refer to ''V'' itself as the representation when the homomorphism is clear from the context.
In the case where ''V'' is of finite dimension ''n'' it is common to choose a
basis for ''V'' and identify GL(''V'') with , the group of ''n''-by-''n''
invertible matrices on the field ''K''.
* If ''G'' is a
topological group and ''V'' is a
topological vector space, a continuous representation of ''G'' on ''V'' is a representation ''ρ'' such that the application defined by is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
.
* The kernel of a representation ''ρ'' of a group ''G'' is defined as the normal subgroup of ''G'' whose image under ''ρ'' is the identity transformation:
::
: A
faithful representation is one in which the homomorphism is
injective; in other words, one whose kernel is the trivial subgroup consisting only of the group's identity element.
* Given two ''K'' vector spaces ''V'' and ''W'', two representations and are said to be equivalent or isomorphic if there exists a vector space
isomorphism so that for all ''g'' in ''G'',
::
Examples
Consider the complex number ''u'' = e
2πi / 3 which has the property ''u''
3 = 1. The set ''C''
3 = forms a
cyclic group under multiplication. This group has a representation ρ on
given by:
:
This representation is faithful because ρ is a
one-to-one map.
Another representation for ''C''
3 on
, isomorphic to the previous one, is σ given by:
:
The group ''C''
3 may also be faithfully represented on
by τ given by:
:
where
:
Another example:
Let
be the space of homogeneous degree-3 polynomials over the complex numbers in variables
Then
acts on
by permutation of the three variables.
For instance,
sends
to
.
Reducibility
A subspace ''W'' of ''V'' that is invariant under the
group action is called a ''
subrepresentation''. If ''V'' has exactly two subrepresentations, namely the zero-dimensional subspace and ''V'' itself, then the representation is said to be irreducible; if it has a proper subrepresentation of nonzero dimension, the representation is said to be reducible. The representation of dimension zero is considered to be neither reducible nor irreducible,
just as the number 1 is considered to be neither
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials ...
nor
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
.
Under the assumption that the
characteristic of the field ''K'' does not divide the size of the group, representations of
finite groups can be decomposed into a
direct sum of irreducible subrepresentations (see
Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
). This holds in particular for any representation of a finite group over the
complex numbers, since the characteristic of the complex numbers is zero, which never divides the size of a group.
In the example above, the first two representations given (ρ and σ) are both decomposable into two 1-dimensional subrepresentations (given by span and span), while the third representation (τ) is irreducible.
Generalizations
Set-theoretical representations
A ''set-theoretic representation'' (also known as a group action or ''permutation representation'') of a
group ''G'' on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''X'' is given by a
function ρ : ''G'' → ''X''
''X'', the set of functions from ''X'' to ''X'', such that for all ''g''
1, ''g''
2 in ''G'' and all ''x'' in ''X'':
:
:
where_
_is_the_identity_element_of_''G''._This_condition_and_the_axioms_for_a_group_imply_that_ρ(''g'')_is_a_bijection.html" ;"title=".html" ;"title="rho(g_2)[x">rho(g_2)[x,
where
is the identity element of ''G''. This condition and the axioms for a group imply that ρ(''g'') is a bijection">.html" ;"title="rho(g_2)[x">rho(g_2)[x,
where
is the identity element of ''G''. This condition and the axioms for a group imply that ρ(''g'') is a bijection (or permutation) for all ''g'' in ''G''. Thus we may equivalently define a permutation representation to be a
group homomorphism from G to the
symmetric group S
''X'' of ''X''.
For more information on this topic see the article on
group action.
Representations in other categories
Every group ''G'' can be viewed as a
category with a single object;
morphisms in this category are just the elements of ''G''. Given an arbitrary category ''C'', a ''representation'' of ''G'' in ''C'' is a
functor from ''G'' to ''C''. Such a functor selects an object ''X'' in ''C'' and a group homomorphism from ''G'' to Aut(''X''), the
automorphism group of ''X''.
In the case where ''C'' is Vect
''K'', the
category of vector spaces over a field ''K'', this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of ''G'' in the
category of sets.
When ''C'' is Ab, the
category of abelian groups, the objects obtained are called
''G''-modules.
For another example consider the
category of topological spaces, Top. Representations in Top are homomorphisms from ''G'' to the
homeomorphism group of a topological space ''X''.
Two types of representations closely related to linear representations are:
*
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group
\mathrm(V) = \mathrm(V) / F^*,
where ...
s: in the category of
projective spaces. These can be described as "linear representations
up to scalar transformations".
*
affine representations: in the category of
affine spaces. For example, the
Euclidean group acts affinely upon
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
.
See also
*
Irreducible representations
*
Character table
*
Character theory
*
Molecular symmetry
*
List of harmonic analysis topics
This is a list of harmonic analysis topics. See also list of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards the classical Fourier series and Fourier transform of mathematical analysis, mathematica ...
*
List of representation theory topics
This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics, which is more directed towards the mathematical analysis aspects of representation theory.
See also: Glossary of representation theory
...
*
Representation theory of finite groups
*
Semisimple representation
In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also calle ...
Notes
References
* . Introduction to representation theory with emphasis on
Lie groups.
* Yurii I. Lyubich.
Introduction to the Theory of Banach Representations of Groups'. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.
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Group theory
Representation theory