Restricted representation

In group theory, restriction forms a representation of a subgroup using a known representation of the whole 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 multiplets, as in the 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.^[1] Restriction can be generalized to other group homomorphisms and to other rings.

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

${\displaystyle \rho \,{\Big |}_{H}}$

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

${\displaystyle \rho \,{\Big |}_{H}(h)=\rho (h).}$