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 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.
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 The results are usually expressed graphically using Young diagrams to encode the signatures used classically to label irreducible representations, familiar from classical invariant theory.
[2][3] 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.
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.
The unitary group U(N) has irreducible representations labelled by signatures 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 The branching rule from U(N) to U(N – 1) states that Example.
Realizing quaternions as 2 x 2 complex matrices, the group Sp(N) is just the group of block matrices (qij) in SU(2N) with where αij and βij are complex numbers.
The irreducible representations of Sp(N) are labelled by signatures where the fi are integers.
The character of the corresponding irreducible representation σf is given by[9] The branching rule from Sp(N) to Sp(N – 1) states that[10] Here fN + 1 = 0 and the multiplicity m(f, g) is given by where is the non-increasing rearrangement of the 2N non-negative integers (fi), (gj) and 0.
The branching from U(2N) to Sp(N) relies on two identities of Littlewood:[11][12][13][14] where Πf,0 is the irreducible representation of U(2N) with signature f1 ≥ ··· ≥ fN ≥ 0 ≥ ··· ≥ 0. where fi ≥ 0.
The branching rule from U(2N) to Sp(N) is given by where all the signature are non-negative and the coefficient M (g, h; k) is the multiplicity of the irreducible representation πk of U(N) in the tensor product πg
In fact if an orthogonal matrix U has eigenvalues zi±1 for 1 ≤ i ≤ n, then the character of the corresponding irreducible representation πf is given by for N = 2n and by for N = 2n+1.
The branching rules from SO(N) to SO(N – 1) state that[17] for N = 2n + 1 and for N = 2n, where the differences fi − gi must be integers.
In this way Gelfand and Tsetlin were able to obtain a basis of any irreducible representation of
Explicit formulas for the action of the Lie algebra on the Gelfand–Tsetlin basis are given in Želobenko (1973).
, a Hopf algebra introduced by Ludwig Faddeev and collaborators, acts irreducibly on this multiplicity space, a fact which enabled Molev (2006) to extend the construction of Gelfand–Tsetlin bases to
[18] In 1937 Alfred H. Clifford proved the following result on the restriction of finite-dimensional irreducible representations from a group G to a normal subgroup N of finite index:[19] Theorem.
Then the restriction of π to N breaks up into a direct sum of irreducible representations of N of equal dimensions.
These irreducible representations of N lie in one orbit for the action of G by conjugation on the equivalence classes of irreducible representations of N. In particular the number of distinct summands is no greater than the index of N in G. Twenty years later George Mackey found a more precise version of this result for the restriction of irreducible unitary representations of locally compact groups to closed normal subgroups in what has become known as the "Mackey machine" or "Mackey normal subgroup analysis".
[20] From the point of view of category theory, restriction is an instance of a forgetful functor.
The relation between restriction and induction in various contexts is called the Frobenius reciprocity.
Taken together, the operations of induction and restriction form a powerful set of tools for analyzing representations.