Editing Restricted representation (section)

In [[group theory]], '''restriction''' forms a [[Group representation|representation]] of a [[subgroup]] using a known representation of the whole [[group (mathematics)|group]]. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an [[irreducible representation]] into irreducible representations of the subgroup are called '''branching rules''', and have important applications in [[physics]]. For example, in case of [[explicit symmetry breaking]], the [[symmetry group]] of the problem is reduced from the whole group to one of its subgroups. In [[quantum mechanics]], this reduction in symmetry appears as a splitting of [[degenerate energy levels]] into [[multiplet]]s, as in the [[Stark effect|Stark]] or [[Zeeman effect]].

The [[induced representation]] is a related operation that forms a representation of the whole group from a representation of a subgroup.  The relation between restriction and induction is described by [[Frobenius reciprocity]] and the Mackey theorem.  Restriction to a [[normal subgroup]] behaves particularly well and is often called [[Clifford theory]] after the theorem of A. H. Clifford.<ref>{{harvnb|Weyl|1946|pp=159–160}}.</ref> Restriction can be generalized to other [[group homomorphism]]s and to other [[ring (mathematics)|rings]].

For any group ''G'', its [[subgroup]] ''H'', and a [[linear representation]] ''ρ'' of ''G'', the restriction of ''ρ'' to ''H'', denoted

: <math> \rho \, \Big|_H </math> 

is a representation of ''H'' on the same [[vector space]] by the same operators:

: <math> \rho\,\Big|_H(h) = \rho(h). </math>