In the

^{2πi / 3} which has the property ''u''^{3} = 1. The _{3} = has a representation ρ on $\backslash mathbb^2$ given by:
:$\backslash rho\; \backslash left(\; 1\; \backslash right)\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash \backslash \; \backslash end\; \backslash qquad\; \backslash rho\; \backslash left(\; u\; \backslash right)\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; u\; \backslash \backslash \; \backslash end\; \backslash qquad\; \backslash rho\; \backslash left(\; u^2\; \backslash right)\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; u^2\; \backslash \backslash \; \backslash end.$
This representation is faithful because ρ is a one-to-one map.
Another representation for ''C''_{3} on $\backslash mathbb^2$, isomorphic to the previous one, is σ given by:
:$\backslash sigma\; \backslash left(\; 1\; \backslash right)\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash \backslash \; \backslash end\; \backslash qquad\; \backslash sigma\; \backslash left(\; u\; \backslash right)\; =\; \backslash begin\; u\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash \backslash \; \backslash end\; \backslash qquad\; \backslash sigma\; \backslash left(\; u^2\; \backslash right)\; =\; \backslash begin\; u^2\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash \backslash \; \backslash end.$
The group ''C''_{3} may also be faithfully represented on $\backslash mathbb^2$ by τ given by:
:$\backslash tau\; \backslash left(\; 1\; \backslash right)\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash \backslash \; \backslash end\; \backslash qquad\; \backslash tau\; \backslash left(\; u\; \backslash right)\; =\; \backslash begin\; a\; \&\; -b\; \backslash \backslash \; b\; \&\; a\; \backslash \backslash \; \backslash end\; \backslash qquad\; \backslash tau\; \backslash left(\; u^2\; \backslash right)\; =\; \backslash begin\; a\; \&\; b\; \backslash \backslash \; -b\; \&\; a\; \backslash \backslash \; \backslash end$
where
:$a=\backslash text(u)=-\backslash tfrac,\; \backslash qquad\; b=\backslash text(u)=\backslash tfrac.$
Another example:
Let $V$ be the space of homogeneous degree-3 polynomials over the complex numbers in variables $x\_1,\; x\_2,\; x\_3.$
Then $S\_3$ acts on $V$ by permutation of the three variables.
For instance, $(12)$ sends $x\_^3$ to $x\_^3$.

^{''X''}, the set of functions from ''X'' to ''X'', such that for all ''g''_{1}, ''g''_{2} in ''G'' and all ''x'' in ''X'':
:$\backslash rho(1);\; href="/html/ALL/s/.html"\; ;"title="">$
:$\backslash rho(g\_1\; g\_2);\; href="/html/ALL/s/.html"\; ;"title="">$
where_$1$_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 $1$ 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 $1$ 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 _{''X''} of ''X''.
For more information on this topic see the article on

_{''K''}, the

Introduction to the Theory of Banach Representations of Groups

'. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988. {{Authority control Group theory Representation theory

mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

field of representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

, group representations describe abstract groups
A group is a number of people 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 identi ...

in terms of bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear transformation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s (i.e. automorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s) of vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s; in particular, they can be used to represent group elements as invertible matrices
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 t ...

so that the group operation can be represented by matrix multiplication
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. Representations of groups are important because they allow many group-theoretic
Image:Rubik's cube.svg, The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups. See Rubik's Cube group.
In mathematics and abstract algebra, group theory studies the algebraic stru ...

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 matrix (mat ...

, which is well understood. They are also important in physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

because, for example, they describe how the symmetry group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...

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
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

from the group to the automorphism group
In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...

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 group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s'' — 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
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "cold drop, frozen drop" ...

and to geometry. If the 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 grassl ...

of scalars of the vector space has characteristic
Characteristic (from the Greek word for a property, attribute or trait
Trait may refer to:
* Phenotypic trait in biology, which involve genes and characteristics of organisms
* Trait (computer programming), a model for structuring object-oriented ...

''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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chan ...

''; this special case has very different properties. See Representation theory of finite groups
The representation theory of group (mathematics), groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on Group action (mathematics), operations of groups on vector spaces. Nevertheless, ...

.
*''Compact group
of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...

s or locally compact group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s'' — 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 In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This Measure (mathematics), measure was introduced by Alfré ...

. The resulting theory is a central part of harmonic analysis
Harmonic analysis is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...

. The Pontryagin duality
300px, The p-adic integer, 2-adic integers, with selected corresponding characters on Prüfer group, their Pontryagin dual group
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that a ...

describes the theory for commutative groups, as a generalised Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

. See also: Peter–Weyl theorem.
*''Lie groups
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

'' — 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s'' (or more generally ''affine group scheme
In mathematics, a group scheme is a type of Algebraic geometry, algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in the se ...

s'') — 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
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, where the relatively weak Zariski topology
In algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained () ...

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 product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, Banach space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

, etc.).
One must also consider the type of 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 grassl ...

over which the vector space is defined. The most important case is the field of complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s. The other important cases are the field of real numbers
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, finite field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, and fields of p-adic number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real number, real and complex number systems. ...

s. In general, algebraically closed
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

fields are easier to handle than non-algebraically closed ones. The characteristic
Characteristic (from the Greek word for a property, attribute or trait
Trait may refer to:
* Phenotypic trait in biology, which involve genes and characteristics of organisms
* Trait (computer programming), a model for structuring object-oriented ...

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 agroup
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

''G'' on a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

''V'' 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 grassl ...

''K'' is a group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from ''G'' to GL(''V''), the general linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

on ''V''. That is, a representation is a map
:$\backslash rho\; \backslash colon\; G\; \backslash to\; \backslash mathrm(V)$
such that
:$\backslash rho(g\_1\; g\_2)\; =\; \backslash rho(g\_1)\; \backslash rho(g\_2)\; ,\; \backslash qquad\; \backslash textg\_1,g\_2\; \backslash in\; G.$
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
Basis may refer to:
Finance and accounting
*Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items.
Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...

for ''V'' and identify GL(''V'') with , the group of ''n''-by-''n'' invertible matrices
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 t ...

on the field ''K''.
* If ''G'' is a topological group and ''V'' is a topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space (an Abstra ...

, 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 ga ...

.
* 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:
::$\backslash ker\; \backslash rho\; =\; \backslash left\backslash .$
: A faithful representationIn mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group (mathematics), group on a vector space is a linear representation in which different elements of are represented by ...

is one in which the homomorphism is injective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

; 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

so that for all ''g'' in ''G'',
::$\backslash alpha\; \backslash circ\; \backslash rho(g)\; \backslash circ\; \backslash alpha^\; =\; \backslash pi(g).$
Examples

Consider the complex number ''u'' = ecyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

''C''Reducibility

A subspace ''W'' of ''V'' that is invariant under thegroup action
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is called a ''subrepresentationIn representation theory, a subrepresentation of a group representation, representation (\pi, V) of a group ''G'' is a representation (\pi, _W, W) such that ''W'' is a vector subspace of ''V'' and \pi, _W(g) = \pi(g), _W.
A finite-dimensional repres ...

''. 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 like 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 (mathematics), 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 ...

.
Under the assumption that the characteristic
Characteristic (from the Greek word for a property, attribute or trait
Trait may refer to:
* Phenotypic trait in biology, which involve genes and characteristics of organisms
* Trait (computer programming), a model for structuring object-oriented ...

of the field ''K'' does not divide the size of the group, representations of finite group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

s can be decomposed into a direct sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

of irreducible subrepresentations (see Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation
In the mathematical field of representation theory
Representation theory is a branch of mathematics
Mathematics (from Ancient Greek, ...

). This holds in particular for any representation of a finite group over the complex numbers
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, 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 agroup
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

''G'' on a set ''X'' is given by a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

ρ : ''G'' → ''X''group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from G to the symmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

Sgroup action
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

.
Representations in other categories

Every group ''G'' can be viewed as acategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

with a single object; morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s in this category are just the elements of ''G''. Given an arbitrary category ''C'', a ''representation'' of ''G'' in ''C'' is a functor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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
In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...

of ''X''.
In the case where ''C'' is Vectcategory of vector spacesIn algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...

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 In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
When ''C'' is Ab, the category of abelian groupsIn mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...

, the objects obtained are called ''G''-modules.
For another example consider the category of topological spaces In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

, Top. Representations in Top are homomorphisms from ''G'' to the homeomorphism
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...

group of a topological space ''X''.
Two types of representations closely related to linear representations are:
*projective representationIn the field of representation theory in mathematics, a projective representation of a group (mathematics), group ''G'' on a vector space ''V'' over a field (mathematics), field ''F'' is a group homomorphism from ''G'' to the projective linear group ...

s: in the category of projective space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s. These can be described as "linear representations up to Two mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

scalar transformations".
*affine representation In mathematics, an affine representation of a topological group, topological Lie group ''G'' on an affine space ''A'' is a continuity (topology), continuous (smooth function, smooth) group homomorphism from ''G'' to the automorphism group of ''A'', ...

s: in the category of affine space
In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...

s. For example, the Euclidean group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

acts affinely upon Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

.
See also

*Irreducible representations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

*Character tableIn group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of character t ...

*Character theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

*Molecular symmetry
Molecular symmetry in chemistry
Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes th ...

* List of harmonic analysis topics
* List of representation theory topics
*Representation theory of finite groups
The representation theory of group (mathematics), groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on Group action (mathematics), operations of groups on vector spaces. Nevertheless, ...

*Semisimple representationIn mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group (mathematics), group or an algebra over a field, algebra that is a direct sum ...

Notes

References

* . Introduction to representation theory with emphasis onLie groups
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

.
* Yurii I. Lyubich. Introduction to the Theory of Banach Representations of Groups

'. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988. {{Authority control Group theory Representation theory