Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication.

Not to be confused with Presentation of a group.

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

Representations of groups allow many 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.

— 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 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. See Representation theory of finite groups.

Finite groups

or locally 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 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.

Compact 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. See Representations of Lie groups and Representations of Lie algebras.

Lie groups

(or more generally affine group schemes) — 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.

Linear algebraic groups

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 be classified in the same way. The general theory for Lie groups deals with of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods.

semidirect products

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.

If G is a and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application Φ : G × V → V defined by Φ(g, v) = ρ(g)(v) is continuous.

topological group

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:

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 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 for V and identify GL(V) with GL(n, K), the group of $n\times n$ invertible matrices on the field K.

Generalizations[edit]

Set-theoretical representations[edit]

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 : G → X^X, the set of functions from X to X, such that for all g₁, g₂ in G and all x in X:

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

; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.. Introduction to representation theory with emphasis on Lie groups.

Fulton, William

Yurii I. Lyubich. . Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.