Editing Group theory (section)

===Representation of groups===
{{Main|Representation theory}}
Saying that a group ''G'' ''[[Group action (mathematics)|acts]]'' on a set ''X'' means that every element of ''G'' defines a bijective map on the set ''X'' in a way compatible with the group structure. When ''X'' has more structure, it is useful to restrict this notion further: a representation of ''G'' on a [[vector space]] ''V'' is a [[group homomorphism]]:

:<math>\rho:G \to \operatorname{GL}(V),</math>

where [[general linear group|GL]](''V'') consists of the invertible [[linear map|linear transformations]] of ''V''. In other words, to every group element ''g'' is assigned an [[automorphism]] ''ρ''(''g'') such that {{nowrap|1=''ρ''(''g'') ∘ ''ρ''(''h'') = ''ρ''(''gh'')}} for any ''h'' in ''G''.

This definition can be understood in two directions, both of which give rise to whole new domains of mathematics.<ref>Such as [[group cohomology]] or [[Equivariant algebraic K-theory|equivariant K-theory]].</ref> On the one hand, it may yield new information about the group ''G'': often, the group operation in ''G'' is abstractly given, but via ''ρ'', it corresponds to the [[matrix multiplication|multiplication of matrices]], which is very explicit.<ref>In particular, if the representation is [[faithful representation|faithful]].</ref> On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if ''G'' is finite, it is known that ''V'' above decomposes into [[irreducible representation|irreducible parts]] (see [[Maschke's theorem]]). These parts, in turn, are much more easily manageable than the whole ''V'' (via [[Schur's lemma]]).

Given a group ''G'', [[representation theory]] then asks what representations of ''G'' exist. There are several settings, and the employed methods and obtained results are rather different in every case: [[representation theory of finite groups]] and representations of [[Lie group]]s are two main subdomains of the theory. The totality of representations is governed by the group's [[character theory|characters]]. For example, [[Fourier series|Fourier polynomial]]s can be interpreted as the characters of [[unitary group|U(1)]], the group of [[complex numbers]] of [[absolute value]] ''1'', acting on the [[Lp space|''L''<sup>2</sup>]]-space of periodic functions.