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. 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
ρ | H {\displaystyle \rho \,{\Big |}_{H}}
is a representation of H on the same vector space by the same operators:
ρ | H ( h ) = ρ ( h ) . {\displaystyle \rho \,{\Big |}_{H}(h)=\rho (h).}
Classical branching rules Classical branching rules describe the restriction of an irreducible complex representation (π, V) of a classical group G to a classical subgroup H, i.e. the multiplicity with which an irreducible representation (σ, W) of H occurs in π. By Frobenius reciprocity for compact groups, this is equivalent to finding the multiplicity of π in the unitary representation induced from σ. Branching rules for the classical groups were determined by
Weyl (1946) between successive unitary groups; Murnaghan (1938) between successive special orthogonal groups and unitary symplectic groups; Littlewood (1950) from the unitary groups to the unitary symplectic groups and special orthogonal groups. The results are usually expressed graphically using Young diagrams to encode the signatures used classically to label irreducible representations, familiar from classical invariant theory. Hermann Weyl and Richard Brauer discovered a systematic method for determining the branching rule when the groups G and H share a common maximal torus: in this case the Weyl group of H is a subgroup of that of G, so that the rule can be deduced from the Weyl character formula. A systematic modern interpretation has been given by Howe (1995) in the context of his theory of dual pairs. The special case where σ is the trivial representation of H was first used extensively by Hua in his work on the Szegő kernels of bounded symmetric domains in several complex variables, where the Shilov boundary has the form G/H. More generally the Cartan-Helgason theorem gives the decomposition when G/H is a compact symmetric space, in which case all multiplicities are one; a generalization to arbitrary σ has since been obtained by Kostant (2004). Similar geometric considerations have also been used by Knapp (2003) to rederive Littlewood's rules, which involve the celebrated Littlewood–Richardson rules for tensoring irreducible representations of the unitary groups. Littelmann (1995) has found generalizations of these rules to arbitrary compact semisimple Lie groups, using his path model, an approach to representation theory close in spirit to the theory of crystal bases of Lusztig and Kashiwara. His methods yield branching rules for restrictions to subgroups containing a maximal torus. The study of branching rules is important in classical invariant theory and its modern counterpart, algebraic combinatorics. Example. The unitary group U(N) has irreducible representations labelled by signatures
f : f 1 ≥ f 2 ≥ ⋯ ≥ f N {\displaystyle \mathbf {f} \,\colon \,f_{1}\geq f_{2}\geq \cdots \geq f_{N}}
where the fi are integers. In fact if a unitary matrix U has eigenvalues zi, then the character of the corresponding irreducible representation πf is given by
Tr π f ( U ) = det z j f i + N − i ∏ i < j ( z i − z j ) . {\displaystyle \operatorname {Tr} \pi _{\mathbf {f} }(U)={\det z_{j}^{f_{i}+N-i} \over \prod _{i<j}(z_{i}-z_{j})}.}
The branching rule from U(N) to U(N – 1) states that
Example. The unitary symplectic group or quaternionic unitary group, denoted Sp(N) or U(N, H), is the group of all transformations of HN which commute with right multiplication by the quaternions H and preserve the H-valued hermitian inner product
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