Editing Restricted representation (section)

== Clifford's theorem ==
{{main|Clifford theory}}
In 1937 [[Alfred H. Clifford]] proved the following result on the restriction of finite-dimensional irreducible representations from a group ''G'' to a normal subgroup ''N'' of finite [[Index of a subgroup|index]]:<ref>{{harvnb|Weyl|1946|pp=159–160, 311}}</ref>

'''Theorem'''. Let {{pi}}: ''G'' <math>\rightarrow </math> GL(''n'',''K'') be an irreducible representation with ''K'' a [[field (mathematics)|field]]. Then 
the restriction of {{pi}} to ''N'' breaks up into a direct sum of irreducible representations of ''N'' of equal dimensions. These irreducible representations of ''N'' lie in one orbit for the action of ''G'' by conjugation on the equivalence classes of irreducible representations of ''N''. In particular the number of distinct summands is no greater than the index of ''N'' in&nbsp;''G''.

Twenty years later [[George Mackey]] found a more precise version of this result for the restriction of irreducible [[unitary representation]]s of [[locally compact group]]s to closed normal subgroups in what has become known as the "Mackey machine" or "Mackey normal subgroup analysis".<ref>{{citation|first=George W.|last=Mackey|authorlink=George Mackey|title=The theory of unitary group representations|series=Chicago Lectures in Mathematics|year=1976|isbn=978-0-226-50052-2}}</ref>