Editing Group representation

{{Short description|Group homomorphism into the general linear group over a vector space}}
{{Distinguish|Presentation of a group}}

[[File:Hexagon_Reflections.png|thumb|right|250px|A representation of a [[group (mathematics)|group]] "acts" on an object. A simple example is how the [[Dihedral group|symmetries of a regular polygon]], consisting of reflections and rotations, transform the polygon.]]

In the [[mathematics|mathematical]] field of [[representation theory]], '''group representations''' describe abstract [[group (mathematics)|groups]] in terms of [[bijective]] [[linear transformation]]s of a [[vector space]] to itself (i.e. vector space [[automorphism]]s); in particular, they can be used to represent group elements as [[invertible matrix|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 allow many [[group theory|group-theoretic]] problems to be reduced to problems in [[linear algebra]]. In [[physics]], 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 group]]s'' &mdash; 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 (mathematics)|field]] of scalars of the vector space has [[characteristic (algebra)|characteristic]] ''p'', and if ''p'' divides the order of the group, then this is called ''[[modular representation theory]]''; this special case has very different properties. See [[Representation theory of finite groups]].
*''[[Compact group]]s or [[locally compact group]]s'' &mdash; 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 the [[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]]'' &mdash; 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]]s'' (or more generally  ''affine [[group scheme]]s'') &mdash; 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'' &mdash; 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 group]]s'' 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]]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]] 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 (mathematics)|field]] over which the vector space is defined. The most important case is the field of [[complex number]]s. The other important cases are the field of [[real numbers]], [[finite field]]s, and fields of [[p-adic number]]s. In general, [[algebraically closed]] fields are easier to handle than non-algebraically closed ones. The [[characteristic (algebra)|characteristic]] of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the [[Order (group theory)|order of the group]].

==Definitions==
A '''representation''' of a [[group (mathematics)|group]] ''G'' on a [[vector space]] ''V'' over a [[field (mathematics)|field]] ''K'' is a [[group homomorphism]] from ''G'' to GL(''V''), the [[general linear group#General linear group of a vector space|general linear group]] on ''V''.  That is, a representation is a map 
:<math>\rho \colon G \to \mathrm{GL}\left(V \right)</math> 
such that
:<math>\rho(g_1 g_2) = \rho(g_1) \rho(g_2) , \qquad \text{for all }g_1,g_2 \in G.</math>

Here ''V'' is called the '''representation space''' and the dimension of ''V'' is called the '''dimension''' or '''degree''' 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 (linear algebra)|basis]] for ''V'' and identify GL(''V'') with {{nowrap|GL(''n'', ''K'')}}, the group of <math>n \times n</math> [[invertible matrix|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 {{nowrap|Φ : ''G'' × ''V'' → ''V''}} defined by {{nowrap|1=Φ(''g'', ''v'') = ''ρ''(''g'')(''v'')}} is [[continuous function (topology)|continuous]].
* 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:
::<math>\ker \rho = \left\{g \in G \mid \rho(g) = \mathrm{id}\right\}.</math>

: A [[faithful representation]] is one in which the homomorphism {{nowrap|''G'' → GL(''V'')}} is [[injective]]; in other words, one whose kernel is the trivial subgroup {''e''} consisting only of the group's identity element.

* Given two ''K'' vector spaces ''V'' and ''W'', two representations {{nowrap|''ρ'' : ''G'' → GL(''V'')}} and {{nowrap|''π'' : ''G'' → GL(''W'')}} are said to be '''equivalent''' or '''isomorphic''' if there exists a vector space [[isomorphism]] {{nowrap|''α'' : ''V'' → ''W''}} so that for all ''g'' in ''G'',
::<math>\alpha \circ \rho(g) \circ \alpha^{-1} = \pi(g).</math>

== Examples ==
Consider the complex number ''u'' = e<sup>2πi / 3</sup> which has the property ''u''<sup>3</sup> = 1. The set ''C''<sub>3</sub> = {1, ''u'', ''u''<sup>2</sup>} forms a [[cyclic group]] under multiplication. This group has a representation ρ on <math>\mathbb{C}^2</math> given by:

:<math>
\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}.
</math>

This representation is faithful because ρ is a [[injective|one-to-one map]].

Another representation for ''C''<sub>3</sub> on <math>\mathbb{C}^2</math>, isomorphic to the previous one, is σ given by:

:<math>
\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}.
</math>

The group ''C''<sub>3</sub> may also be faithfully represented on <math>\mathbb{R}^2</math> by τ given by:

:<math>
\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}
</math>

where

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

A possible representation on <math>\mathbb{R}^3</math> is given by the set of cyclic permutation matrices ''v'':

:<math>
\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}
.</math>

Another example:

Let <math>V</math> be the space of homogeneous degree-3 polynomials over the complex numbers in variables <math>x_1, x_2, x_3. </math>

Then <math>S_3</math> acts on <math>V</math> by permutation of the three variables.

For instance, <math>(12)</math> sends <math>x_{1}^3</math> to <math>x_{2}^3</math>.

== Reducibility ==

{{main|Irreducible representation}}

A subspace ''W'' of ''V'' that is invariant under the [[Group action (mathematics)|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, <ref>{{Cite web|date=2019-09-04|title=1.4: Representations|url=https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_and_Websites_(Inorganic_Chemistry)/Advanced_Inorganic_Chemistry_(Wikibook)/01%3A_Chapters/1.04%3A_Representations|access-date=2021-06-23|website=Chemistry LibreTexts|language=en}}</ref> just as the number 1 is considered to be neither [[Composite number|composite]] nor [[Prime number|prime]].

Under the assumption that the [[characteristic (algebra)|characteristic]] of the field ''K'' does not divide the size of the group, representations of [[finite group]]s can be decomposed into a [[direct sum of groups|direct sum]] of irreducible subrepresentations (see [[Maschke's theorem]]). 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{(1,0)} and span{(0,1)}), 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 (mathematics)|group]] ''G'' on a [[Set (mathematics)|set]] ''X'' is given by a [[function (mathematics)|function]] ρ : ''G'' → ''X''<sup>''X''</sup>, the set of functions from ''X'' to ''X'', such that for all ''g''<sub>1</sub>, ''g''<sub>2</sub> in ''G'' and all ''x'' in ''X'':

:<math>\rho(1)[x] = x</math>
:<math>\rho(g_1 g_2)[x]=\rho(g_1)[\rho(g_2)[x]],</math>

where <math>1</math> 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<sub>''X''</sub> of ''X''.

For more information on this topic see the article on [[Group action (mathematics)|group action]].

===Representations in other categories===
Every group ''G'' can be viewed as a [[category (mathematics)|category]] with a single object; [[morphism]]s 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'''<sub>''K''</sub>, 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-module|''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]]s: in the category of [[projective space]]s. These can be described as "linear representations [[up to]] scalar transformations". 
*[[affine representation]]s: in the category of [[affine space]]s. For example, the [[Euclidean group]] acts affinely upon [[Euclidean space]].

==See also==
*[[Irreducible representations]]
*[[Character table]]
*[[Character theory]]
*[[Molecular symmetry]]
*[[List of harmonic analysis topics]]
*[[List of representation theory topics]]
*[[Representation theory of finite groups]]
*[[Semisimple representation]]

==Notes==
{{reflist}}

==References==
* {{Fulton-Harris}}. Introduction to representation theory with emphasis on [[Lie groups]].
* Yurii I. Lyubich. ''[https://books.google.com/books?id=gCr3BwAAQBAJ&q=%22Introduction+to+the+Theory+of+Banach+Representations+of+Groups%22 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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[[Category:Group theory]]
[[Category:Representation theory]]
[[Category:Representation theory of groups| ]]