Group representation

Last updated
A representation of a group "acts" on an object. A simple example is how the symmetries of a regular polygon, consisting of reflections and rotations, transform the polygon. Hexagon Reflections.png
A representation of a group "acts" on an object. A simple example is how the symmetries of a regular polygon, consisting of reflections and rotations, transform the polygon.

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations (i.e. automorphisms) of vector spaces; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.

Contents

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:

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 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.

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

such that

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.

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 only of the group's identity element.

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 given by:

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

Another representation for C3 on , isomorphic to the previous one, is σ given by:

The group C3 may also be faithfully represented on by τ given by:

where

Another example:

Let be the space of homogeneous degree-3 polynomials over the complex numbers in variables

Then acts on by permutation of the three variables.

For instance, sends to .

Reducibility

A subspace W of V that is invariant 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, [1] 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 groups can be decomposed into a 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 G on a set X is given by a function ρ : GXX, the set of functions from X to X, such that for all g1, g2 in G and all x in X:

where 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 SX 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; 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 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.

When C is Ab, the category of abelian groups, the objects obtained are called 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:

See also

Related Research Articles

Associative algebra Algebraic structure with (a + b)(c + d) = ac + ad + bc + bd and (a)(bc) = (ab)(c)

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.

Lie algebra vector space with an alternating binary operation satisfying the Jacobi identity.

In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

Representation of a Lie group Group representation

In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.

Adjoint representation

In mathematics, the adjoint representation of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: .

Lie algebra representation

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.

In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is one-to-one with the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

Irreducible representation Type of group and algebra representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of .

In mathematics, a Casimir element is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.

In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear transformation from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0. An important special case occurs when M = N and φ is a self-map; in particular, any element of the center of a group must act as a scalar operator on M. The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which is due to Jacques Dixmier.

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representationρ* is defined over the dual vector space V* as follows:

In mathematics, especially in an area of abstract algebra known as representation theory, 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, especially in the area of algebra known as representation theory, the representation ring of a group is a ring formed from all the finite-dimensional linear representations of the group. Elements of the representation ring are sometimes called virtual representations. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of algebraically closed fields of characteristic p where the Sylow p-subgroups are cyclic is also theoretically approachable.

Representation theory Branch of mathematics that studies abstract algebraic structures

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.

In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.

This is a glossary of representation theory in mathematics.

In mathematics, specifically in representation theory, a semisimple representation is a linear representation of a group or an algebra that is a direct sum of simple representations. It is an example of the general mathematical notion of semisimplicity.

References

  1. "1.4: Representations". Chemistry LibreTexts. 2019-09-04. Retrieved 2021-06-23.