Restricted representation

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In 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. Explicit descriptions of restricted representations are often called branching rules, particularly in the completely reducible cases.

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.