Restricted representation

In <a href="/facts/Group_theory/NfowcI5H">group theory</a>, restriction forms a <a href="/facts/Group_representation/FsuuDagQ">representation</a> of a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> using a known representation of the whole <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>. 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 <a href="/facts/Irreducible_representation/Hu5geBgk">irreducible representation</a> into irreducible representations of the subgroup are called branching rules, and have important applications in <a href="/facts/Physics/a7aH2y08">physics</a>. For example, in case of <a href="/facts/Explicit_symmetry_breaking/Y1NetSrN">explicit symmetry breaking</a>, the <a href="/facts/Symmetry_group/CNqeseMp">symmetry group</a> of the problem is reduced from the whole group to one of its subgroups. In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, this reduction in symmetry appears as a splitting of <a href="/facts/Degenerate_energy_levels/ulrnMXTR">degenerate energy levels</a> into <a href="/facts/Multiplet/D8fr0BM8">multiplets</a>, as in the <a href="/facts/Stark_effect/NRLArv1u">Stark</a> or <a href="/facts/Zeeman_effect/A6NMlp2D">Zeeman effect</a>.
The <a href="/facts/Induced_representation/Jh25PGDP">induced representation</a> 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 <a href="/facts/Frobenius_reciprocity/qoLTxRCX">Frobenius reciprocity</a> and the Mackey theorem. Restriction to a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> behaves particularly well and is often called <a href="/facts/Clifford_theory/32sQWx2j">Clifford theory</a> after the theorem of A. H. Clifford. Restriction can be generalized to other <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphisms</a> and to other <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a>.
For any group G, its <a href="/facts/Subgroup/r91mBgpa">subgroup</a> H, and a <a href="/facts/Linear_representation/3ydLtaGH">linear representation</a> ρ of G, the restriction of ρ to H, denoted

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

is a representation of H on the same <a href="/facts/Vector_space/M1pxMLx2">vector space</a> by the same operators:

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

Restricted representation open-in-new

Restricted representation