- Group representation
In the mathematical field of

representation theory ,**group representations**describe abstract groups in terms oflinear transformation s ofvector space s; in particular, they can be used to represent group elements as matrices so that the group operation can be represented bymatrix multiplication . Representations of groups are important because they allow many group-theoretic problems to be reduced to problems inlinear algebra , which is well-understood. They are also important inphysics because, for example, they describe how thesymmetry 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 theautomorphism 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" — 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 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 "; this special case has very different properties. SeeRepresentation theory of finite groups .*"

Compact group s orlocally compact group 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, usingHaar 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 group s" (or more generally "affinegroup scheme 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 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. 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 product s of the two types, by means of general results called "Mackey theory ", which is a generalization ofWigner'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 aHilbert 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 number s. The other important cases are the field ofreal numbers ,finite field s, and fields ofp-adic number s. 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 avector space "V" over a field "K" is agroup homomorphism from "G" to "GL(V)", the general linear group on "V". That is, a representation is a

$ho\; colon\; G\; o\; GL(V)\; ,!$ such that:$ho(g\_1\; g\_2)\; =\; ho(g\_1)\; ho(g\_2)\; ,\; qquad\; ext\{for\; all\; \}g\_1,g\_2\; in\; G\; .\; ,!$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 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 GL ("n", "K") 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 $ho$ such that the application $Phi:G\; imes\; V\; o\; V$ defined by $Phi(g,v)=\; ho(g).v$ is continuous.The

**kernel**of a representation $ho$ of a group "G" is defined as the normal subgroup of "G" whose image under $ho$ is the identity transformation:: $ker\; ho\; =\; left\{g\; in\; G\; mid\; ho(g)\; =\; id\; ight\}\; ,!.$A

faithful representation is one in which the homomorphism "G" → GL("V") isinjective ; in other words, one whose kernel is the trivial subgroup {"e"} consisting of just the group's identity element.Given two "K" vector spaces "V" and "W", two representations:$ho\_1\; colon\; G\; o\; GL(V)\; ,!$ and :$ho\_2\; colon\; G\; ightarrow\; GL(W)\; ,!$are said to be

**equivalent**or**isomorphic**if there exists a vector spaceisomorphism :$alpha\; colon\; V\; o\; W\; ,!$so that for all "g" in "G":$alpha\; circ\; ho\_1(g)\; circ\; alpha^\{-1\}\; =\; ho\_2(g)\; ,!$**Examples**Consider the complex number "u" = e

^{2πi / 3}which has the property "u"^{3}= 1. Thecyclic group "C"_{3}= {1, "u", "u"^{2}} has a representation ρ on**C**^{2}given by::$ho\; left(\; 1\; ight)\; =egin\{bmatrix\}1\; 0\; \backslash 0\; 1\; \backslash end\{bmatrix\}qquad\; ho\; left(\; u\; ight)\; =egin\{bmatrix\}1\; 0\; \backslash 0\; u\; \backslash end\{bmatrix\}qquad\; ho\; left(\; u^2\; ight)\; =egin\{bmatrix\}1\; 0\; \backslash 0\; u^2\; \backslash end\{bmatrix\}$

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

An isomorphic representation for "C"

_{3}is:$ho\; left(\; 1\; ight)\; =egin\{bmatrix\}1\; 0\; \backslash 0\; 1\; \backslash end\{bmatrix\}qquad\; ho\; left(\; u\; ight)\; =egin\{bmatrix\}u\; 0\; \backslash 0\; 1\; \backslash end\{bmatrix\}qquad\; ho\; left(\; u^2\; ight)\; =egin\{bmatrix\}u^2\; 0\; \backslash 0\; 1\; \backslash end\{bmatrix\}$

The group "C"

_{3}may also be faithfully represented on**R**^{2}by:$ho\; left(\; 1\; ight)\; =egin\{bmatrix\}1\; 0\; \backslash 0\; 1\; \backslash end\{bmatrix\}qquad\; ho\; left(\; u\; ight)\; =egin\{bmatrix\}a\; -b\; \backslash b\; a\; \backslash end\{bmatrix\}qquad\; ho\; left(\; u^2\; ight)\; =egin\{bmatrix\}a\; b\; \backslash -b\; a\; \backslash end\{bmatrix\}$where $a=Re(u)=-1/2$ and $b=Im(u)=sqrt\{3\}/2$.

**Reducibility**A subspace "W" of "V" that is fixed 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 like the number 1 is considered to be neither composite nor prime.Under the assumption that the characteristic of the field "K" does not divide the size of the group, representations of

finite group s 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 are both decomposable into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}), while the third representation is irreducible.

**Generalizations****et-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 "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"::$ho(1)\; [x]\; =\; x$:$ho(g\_1\; g\_2)\; [x]\; =\; ho(g\_1)\; [\; ho(g\_2)\; [x]$

This condition and the axioms for a group imply that ρ("g") is a

bijection (orpermutation ) for all "g" in "G". Thus we may equivalently define a permutation representation to be agroup homomorphism from G to thesymmetric 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;

morphism s in this category are just the elements of "G". Given an arbitrary category "C", a "representation" of "G" in "C" is afunctor from "G" to "C". Such a functor selects an object "X" in "C" and a group homomorphism from "G" to Aut("X"), theautomorphism group of "X".In the case where "C" is

**Vect**_{"K"}, thecategory 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 thecategory of sets .For another example consider the

category of topological spaces ,**Top**. Representations in**Top**are homomorphisms from "G" to thehomeomorphism group of a topological space "X".Two types of representations closely related to linear representations are:

*projective representation s: in the category ofprojective space s. These can be described as "linear representationsup to scalar transformations".

*affine representation s: in the category ofaffine space s. For example, theEuclidean group acts affinely uponEuclidean space .**ee also***

Character theory

*List of harmonic analysis topics

*List of representation theory topics

*Representation theory of finite groups **References***. Introduction to representation theory with emphasis on

Lie groups .

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2010.*

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