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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 "description" means a homomorphism from the group to the automorphism group of the object. If the object is a vector space we have a linear representation. Some people use realization for this notion and reserve the term representation for linear representations. The bulk of this article desribes linear representation theory; see the last section for generalizations.

1 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:

• Finite groups — 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 below). See Representation of a finite group.

• Compact or locally compact topological 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 Haar measureIn mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. This measure was introduced by Alfred Haar, a Hungarian. The resulting theory is a central part of harmonic analysisHarmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called ". The Pontryagin dualityTopological groups Harmonic analysis Theorems In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of describes the theory for commutative groups, as a generalised Fourier transformThe Fourier transform named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes").. See also: Peter-Weyl theoremThe Peter-Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Peter, in the setting of a compact Lie.

• 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. See Representations of Lie groupsIn mathematics and theoretical physics, the idea of a representation of a Lie group plays 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 correspondi and Representations of Lie algebras.

• 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 products 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 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 quite significant; many theorems depend on the order of the group not dividing the characteristic of the field. Representations over a finite field are called modular representations.