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Group representation

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From Wikipedia, the free encyclopedia

Group homomorphism into the general linear group over a vector space

Not to be confused withPresentation of a group.

A representation of agroup "acts" on an object. A simple example is how thesymmetries of a regular polygon, consisting of reflections and rotations, transform the polygon.

In themathematical field ofrepresentation theory,group representations describe abstractgroups in terms ofbijective linear transformations of avector space to itself (i.e. vector spaceautomorphisms); in particular, they can be used to represent group elements asinvertible matrices so that the group operation can be represented bymatrix multiplication.

In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.

Representations of groups allow manygroup-theoretic problems to be reduced to problems inlinear algebra. Inphysics, they describe how thesymmetry group of a physical system affects the solutions of equations describing that system.

The termrepresentation 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 ahomomorphism from the group to theautomorphism group of an object. If the object is a vector space we have alinear representation. Some people userealization for the general notion and reserve the termrepresentation 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

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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 tocrystallography and to geometry. If thefield of scalars of the vector space hascharacteristicp, and ifp divides the order of the group, then this is calledmodular representation theory; this special case has very different properties. SeeRepresentation theory of finite groups.
Compact groups orlocally 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 theHaar measure. The resulting theory is a central part ofharmonic analysis. ThePontryagin duality describes the theory for commutative groups, as a generalisedFourier 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. SeeRepresentations of Lie groups andRepresentations of Lie algebras.
Linear algebraic groups (or more generallyaffinegroup schemes) — These are the analogues of Lie groups, but over more general fields than justR orC. 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 fromalgebraic geometry, where the relatively weakZariski 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. Thesemisimple Lie groups have a deep theory, building on the compact case. The complementarysolvable Lie groups cannot be classified in the same way. The general theory for Lie groups deals withsemidirect products of the two types, by means of general results calledMackey theory, which is a generalization ofWigner's classification methods.

Representation theory also depends heavily on the type ofvector 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 aHilbert space,Banach space, etc.).

One must also consider the type offield over which the vector space is defined. The most important case is the field ofcomplex numbers. The other important cases are the field ofreal numbers,finite fields, and fields ofp-adic numbers. In general,algebraically closed fields are easier to handle than non-algebraically closed ones. Thecharacteristic of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing theorder of the group.

Definitions

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Arepresentation of agroupG on avector spaceV over afieldK is agroup homomorphism fromG to GL(V), thegeneral linear group onV. That is, a representation is a map

\rho \colon G\to \mathrm {GL} \left(V\right)

such that

\rho (g_{1}g_{2})=\rho (g_{1})\rho (g_{2}),\qquad {\text{for all }}g_{1},g_{2}\in G.

HereV is called therepresentation space and the dimension ofV is called thedimension ordegree of the representation. It is common practice to refer toV itself as the representation when the homomorphism is clear from the context.

In the case whereV is of finite dimensionn it is common to choose abasis forV and identify GL(V) withGL(n,K), the group of $n\times n$ invertible matrices on the fieldK.

IfG is atopological group andV is atopological vector space, acontinuous representation ofG onV is a representationρ such that the applicationΦ :G ×V →V defined byΦ(g,v) =ρ(g)(v) iscontinuous.
Thekernel of a representationρ of a groupG is defined as the normal subgroup ofG whose image underρ is the identity transformation:

\ker \rho =\left\{g\in G\mid \rho (g)=\mathrm {id} \right\}.

Afaithful representation is one in which the homomorphismG → GL(V) isinjective; in other words, one whose kernel is the trivial subgroup {e} consisting only of the group's identity element.

Given twoK vector spacesV andW, two representationsρ :G → GL(V) andπ :G → GL(W) are said to beequivalent orisomorphic if there exists a vector spaceisomorphismα :V →W so that for allg inG,

\alpha \circ \rho (g)\circ \alpha ^{-1}=\pi (g).

Examples

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Consider the complex numberu = e^{2πi / 3} which has the propertyu³ = 1. The setC₃ = {1,u,u²} forms acyclic group under multiplication. This group has a representation ρ on $\mathbb {C} ^{2}$ given by:

\rho \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \rho \left(u\right)={\begin{bmatrix}1&0\\0&u\\\end{bmatrix}}\qquad \rho \left(u^{2}\right)={\begin{bmatrix}1&0\\0&u^{2}\\\end{bmatrix}}.

This representation is faithful because ρ is aone-to-one map.

Another representation forC₃ on $\mathbb {C} ^{2}$ , isomorphic to the previous one, is σ given by:

\sigma \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \sigma \left(u\right)={\begin{bmatrix}u&0\\0&1\\\end{bmatrix}}\qquad \sigma \left(u^{2}\right)={\begin{bmatrix}u^{2}&0\\0&1\\\end{bmatrix}}.

The groupC₃ may also be faithfully represented on $\mathbb {R} ^{2}$ by τ given by:

\tau \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \tau \left(u\right)={\begin{bmatrix}a&-b\\b&a\\\end{bmatrix}}\qquad \tau \left(u^{2}\right)={\begin{bmatrix}a&b\\-b&a\\\end{bmatrix}}

where

a={\text{Re}}(u)=-{\tfrac {1}{2}},\qquad b={\text{Im}}(u)={\tfrac {\sqrt {3}}{2}}.

A possible representation on $\mathbb {R} ^{3}$ is given by the set of cyclic permutation matricesv:

\upsilon \left(1\right)={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}}\qquad \upsilon \left(u\right)={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\\\end{bmatrix}}\qquad \upsilon \left(u^{2}\right)={\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\\\end{bmatrix}}.

Another example:

Let $V {\displaystyle V}$ 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 $V {\displaystyle V}$ by permutation of the three variables.

For instance, $(12) {\displaystyle (12)}$ sends $x_{1}^{3}$ to $x_{2}^{3}$ .

Reducibility

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Main article:Irreducible representation

A subspaceW ofV that is invariant under thegroup action is called asubrepresentation. IfV has exactly two subrepresentations, namely the zero-dimensional subspace andV itself, then the representation is said to beirreducible; if it has a proper subrepresentation of nonzero dimension, the representation is said to bereducible. The representation of dimension zero is considered to be neither reducible nor irreducible,^[1] just as the number 1 is considered to be neithercomposite norprime.

Under the assumption that thecharacteristic of the fieldK does not divide the size of the group, representations offinite groups can be decomposed into adirect sum of irreducible subrepresentations (seeMaschke's theorem). This holds in particular for any representation of a finite group over thecomplex 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{(1,0)} and span{(0,1)}), while the third representation (τ) is irreducible.

Generalizations

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Set-theoretical representations

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Aset-theoretic representation (also known as a group action orpermutation representation) of agroupG on asetX is given by afunction ρ :G →X^X, the set of functions fromX toX, such that for allg₁,g₂ inG and allx inX:

\rho (1)[x]=x

\rho (g_{1}g_{2})[x]=\rho (g_{1})[\rho (g_{2})[x]],

where $1 {\displaystyle 1}$ is the identity element ofG. This condition and the axioms for a group imply that ρ(g) is abijection (orpermutation) for allg inG. Thus we may equivalently define a permutation representation to be agroup homomorphism from G to thesymmetric group S_X ofX.

For more information on this topic see the article ongroup action.

Representations in other categories

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Every groupG can be viewed as acategory with a single object;morphisms in this category are just the elements ofG. Given an arbitrary categoryC, arepresentation ofG inC is afunctor fromG toC. Such a functor selects an objectX inC and a group homomorphism fromG to Aut(X), theautomorphism group ofX.

In the case whereC isVect_K, thecategory of vector spaces over a fieldK, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation ofG in thecategory of sets.

WhenC isAb, thecategory of abelian groups, the objects obtained are calledG-modules.

For another example consider thecategory of topological spaces,Top. Representations inTop are homomorphisms fromG to thehomeomorphism group of a topological spaceX.

Two types of representations closely related to linear representations are: