Editing Equivariant map (section)

===Representation theory===
{{See also|Representation theory#Equivariant maps and isomorphisms}}

In the [[representation theory of finite groups]], a vector space equipped with a group that acts by linear transformations of the space is called a [[linear representation]] of the group.
A [[linear map]] that commutes with the action is called an '''intertwiner'''. That is, an intertwiner is just an equivariant linear map between two representations. Alternatively, an intertwiner for representations of a group {{mvar|G}} over a [[field (mathematics)|field]] {{mvar|K}} is the same thing as a [[module (mathematics)|module homomorphism]] of {{math|''K''[''G'']}}-[[module (mathematics)|modules]], where {{math|''K''[''G'']}} is the [[group ring]] of ''G''.<ref>{{citation
 | last1 = Fuchs | first1 = Jürgen
 | last2 = Schweigert | first2 = Christoph
 | isbn = 0-521-56001-2
 | mr = 1473220
 | page = 70
 | publisher = Cambridge University Press, Cambridge
 | series = Cambridge Monographs on Mathematical Physics
 | title = Symmetries, Lie algebras and representations: A graduate course for physicists
 | url = https://books.google.com/books?id=B_JQryjNYyAC&pg=PA70
 | year = 1997}}.</ref>

Under some conditions, if ''X'' and ''Y'' are both [[irreducible representation]]s, then an intertwiner (other than the [[zero map]]) only exists if the two representations are equivalent (that is, are [[isomorphic]] as [[module (mathematics)|modules]]). That intertwiner is then unique [[up to]] a multiplicative factor (a non-zero [[scalar (mathematics)|scalar]] from {{mvar|K}}). These properties hold when the image of {{math|''K''[''G'']}} is a simple algebra, with centre {{mvar|K}} (by what is called [[Schur's lemma]]: see [[simple module]]). As a consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same.<ref>{{citation
 | last1 = Sexl | first1 = Roman U.
 | last2 = Urbantke | first2 = Helmuth K.
 | doi = 10.1007/978-3-7091-6234-7
 | isbn = 3-211-83443-5
 | location = Vienna
 | mr = 1798479
 | page = 165
 | publisher = Springer-Verlag
 | series = Springer Physics
 | title = Relativity, groups, particles: Special relativity and relativistic symmetry in field and particle physics
 | url = https://books.google.com/books?id=iyj0CAAAQBAJ&pg=PA165
 | year = 2001}}.</ref>