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Encyclopedia > Group representation

Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. Representation theory is important because it enables many group-theoretic problems to be reduced to problems in linear algebra, which is a very well-understood theory. It is also important in physics because, for example, it is used to describe how the symmetry group of a physical system affects the solutions of equations describing that system. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Group theory is that branch of mathematics concerned with the study of groups. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time and explaining them using mathematics. ... The symmetry group of an object (e. ...


Representations can also be defined for other mathematical structures, such as associative algebras, and Lie or Hopf algebras; for the rest of this article representation and representation theory will refer only to representation of groups. In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...


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 the 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. In abstract algebra, a homomorphism is a structure-preserving map. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

Contents

Branches of representation theory

Representation theory 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:

  • 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 groups (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 in the same way be classified. 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.). In mathematics, a finite group is a group which has finitely many elements. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, modular representation theory is the branch of representation theory that studies linear representations of finite group G over a field K such that the characteristic of K divides the order of G. In other words, the number of elements of G is zero when considered as an element... In mathematics, the general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... The Peter-Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ* on tangent spaces is a homomorphism of Lie algebras, i. ... In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. ... In mathematics, a group scheme is a group object (some would prefer to say just group) in the category of schemes. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... In mathematics and theoretical physics, Wigners classification is a classification of the nonnegative energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...


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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Please refer to Real vs. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... The p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...


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 This picture illustrates how the hours in a clock form a group. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... 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) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ...

rho colon G to GL(V) ,!

such that

rho(g_1 g_2) = rho(g_1) rho(g_2) , qquad text{for all }g_1,g_2 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 context (and, often, even when it is not).


In the case where V is of finite dimension n it is common to choose a basis for V and identify GL(V) with GL (n, K) the group of n-by-n invertible matrices on the field K. In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...


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:

ker rho = left{g in G mid rho(g) = idright} ,!

A faithful representation is one in which the homomorphism G → GL(V) is injective; in other words, one whose kernel is the trivial subgroup {e} consisting of just the group's identity element. In mathematics, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...


Given two K vector spaces V and W, two representations

rho_1 colon G to GL(V) ,!

and

rho_2 colon Grightarrow GL(W) ,!

are said to be equivalent or isomorphic if there exists a vector space isomorphism In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

alpha colon V to W ,!

so that for all g in G

alpha circ rho_1(g) circ alpha^{-1} = rho_2(g) ,!

Examples

Consider the complex number u = e2πi / 3 which has the property u3 = 1. The cyclic group C3 = {1, u, u2} has a representation ρ on C2 given by: In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

begin{bmatrix} 1 & 0  0 & 1  end{bmatrix} qquad begin{bmatrix} 1 & 0  0 & u  end{bmatrix} qquad begin{bmatrix} 1 & 0  0 & u^2  end{bmatrix}

(the three matrices are ρ(1), ρ(u) and ρ(u2) respectively). This representation is faithful because ρ is a one-to-one map. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...


An isomorphic representation for C3 is

begin{bmatrix} 1 & 0  0 & 1  end{bmatrix} qquad begin{bmatrix} u & 0  0 & 1  end{bmatrix} qquad begin{bmatrix} u^2 & 0  0 & 1  end{bmatrix}

Reducibility

A subspace W of V that is fixed under the group action is called a subrepresentation. If V has a non-zero proper subrepresentation, the representation is said to be reducible. Otherwise, it is said to be irreducible. In mathematics, a symmetry group describes all symmetries of objects. ...


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). This is true for representations over the complex numbers. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, a finite group is a group which has finitely many elements. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, in particular group representation theory, Maschkes theorem is the basic result proving that linear representations of a finite group over the complex numbers break up into irreducible pieces. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...


In the example above, the representation given is decomposable into two 1-dimensional subrepresentations (given by span{(1,0) } and span{(0,1)}).


Generalizations

Set-theoretical representations

A set-theoretic representation (also known as a group action or permutation representation) of a group G on a set X is given by a function ρ from G to XX, the set of functions from X to X, such that for all g1, g2 in G and all x in X: In mathematics, a symmetry group describes all symmetries of objects. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Partial plot of a function f. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Partial plot of a function f. ...

ρ(1)[x] = x
ρ(g1g2)[x] = ρ(g1)[ρ(g2)[x]]

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 SX of X. A bijective function. ... Permutation is the arrangement of symbols or objects into distinguishable orderings. ... 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) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...


For more information on this topic see the article on group action. In mathematics, a symmetry group describes all symmetries of objects. ...


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 mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...


In the case where C is VectK, 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. In mathematics, the category K_Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...


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. The category Top has topological spaces as objects and continuous maps as morphisms. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...


Two types of representations closely related to linear representations are:

In mathematics, in particular in group theory, if G is a group and ρ is a vector space over a field K, then a projective representation is a homomorphism from G to Aut(ρ)/Kx where Kx here is the normal subgroup of Aut(ρ) consisting of multiplications of vectors... In mathematics, a projective space is a fundamental construction from any vector space. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... An affine representation of a topological (Lie) group G is a continuous (smooth) homomorphism from G to the automorphism group of an affine space, A. An example is the action of the Euclidean group E(n) upon the Euclidean space En. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

See also


  Results from FactBites:
 
Group representation - Wikipedia, the free encyclopedia (1365 words)
Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces.
Representation theory is important because it enables many group-theoretic problems to be reduced to problems in linear algebra, which is a very well-understood theory.
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.
  More results at FactBites »

 
 

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