Editing Group representation (section)

==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'' &mdash; 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 (mathematics)|field]] of scalars of the vector space has [[characteristic (algebra)|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]].
*''[[Compact group]]s or [[locally compact group]]s'' &mdash; 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]].
*''[[Lie groups]]'' &mdash; 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]].
*''[[Linear algebraic group]]s'' (or more generally  ''affine [[group scheme]]s'') &mdash; 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.
*''Non-compact topological groups'' &mdash; 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 group]]s'' 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 [[semidirect product]]s 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 (mathematics)|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 of [[real numbers]], [[finite field]]s, and fields of [[p-adic number]]s. In general, [[algebraically closed]] fields are easier to handle than non-algebraically closed ones. The [[characteristic (algebra)|characteristic]] of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the [[Order (group theory)|order of the group]].