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"Representation theory" redirects here. For other mathematical representation theorems, see representation theorem. In Mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is Isomorphic to a concrete structure

Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Linear algebra is the branch of Mathematics concerned with 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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is

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 a module) which also allows the multiplication of vectors in a distributive In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( Unital associative algebra, a Coalgebra

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. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself 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.

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:

Representation theory also depends heavily on the type of vector space on which the group acts. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) 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. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division The most important case is the field of complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The other important cases are the field of real numbers, finite fields, and fields of p-adic numbers. In Mathematics, the real numbers may be described informally in several different ways 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 In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In general, algebraically closed fields are easier to handle than non-algebraically closed ones. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients 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 Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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 In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation That is, a representation is a map

\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 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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by-

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 \Phi:G\times V\to V defined by Φ(g,v) = ρ(g). v is continuous. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

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) = id\right\} \,\!.

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, especially in the area of Abstract algebra known as Representation theory, a faithful representation ρ of a group G

Given two K vector spaces V and W, two representations

\rho_1 \colon G \to GL(V) \,\!

and

\rho_2 \colon G\rightarrow GL(W) \,\!

are said to be equivalent or isomorphic if there exists a vector space isomorphism

\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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective The cyclic group C3 = {1, u, u2} has a representation ρ on C2 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 a one-to-one map. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

An isomorphic representation for C3 is

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

The group C3 may also be faithfully represented on R2 by

\rho \left( 1 \right) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u \right) = \begin{bmatrix} a & -b \\ b & a \\ \end{bmatrix} \qquad \rho \left( u^2 \right) = \begin{bmatrix} a & b \\ -b & a \\ \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. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. 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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

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). In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, a finite group is a group which has finitely many elements The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction Maschke's theorem is a theorem in Group representation theory which concerns the decomposition of representations of a Finite group into irreducible pieces 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

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

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:

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. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In several fields of Mathematics the term permutation is used with different but closely related meanings Thus we may equivalently define a permutation representation to be a group homomorphism from G to the symmetric group SX of X. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

For more information on this topic see the article on group action. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

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. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and Given an arbitrary category C, a representation of G in C is a functor from G to C. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories 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, 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. In Mathematics, especially Category theory, the category K-Vect has all Vector spaces over a fixed field K as objects Likewise, a set-theoretic representation is just a representation of G in the category of sets. In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are

For another example consider the category of topological spaces, Top. In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose Representations in Top are homomorphisms from G to the homeomorphism group of a topological space X. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

Two types of representations closely related to linear representations are:

See also

References

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