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Restricted RepresentationIn mathematics, if G is a group and H a subgroup, then for any linear representation ρ of G, we can define the restricted representation - ρ|H
by simply setting - ρ|H(h) = ρ(h).
This rather evident construction may be extended in numerous and significant ways. For instance we may take any group homomorphism φ from H to G, instead of the inclusion map, and define the restricted representation of H by the composition - ρoφ.
We may also apply the idea to other categories in abstract algebra: associative algebras, rings, Lie algebras, Lie superalgebras, Hopf algebras to name some. Representations or modules restrict to subobjects, or via homomorphisms. In a general sense, restriction of representations is a type of forgetful functor, and adjoint to the construction of induced representations or modules. This aspect comes from category theory; its implications are different in the various cases, that of the representation theory of a finite group being rather well-behaved.
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