Classical branching rules describe the restriction of an irreducible
complex representation (,
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 . By Frobenius reciprocity for
compact groups, this is equivalent to finding the multiplicity of in the
unitary representation induced from σ. Branching rules for the classical groups were determined by • between successive
unitary groups; • between successive
special orthogonal groups and
unitary symplectic groups; • 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 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 . Similar geometric considerations have also been used by to rederive Littlewood's rules, which involve the celebrated
Littlewood–Richardson rules for tensoring irreducible representations of the unitary groups. 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 :\mathbf{f} \,\colon \,f_1\ge f_2\ge \cdots \ge 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
f is given by : \operatorname{Tr} \pi_{\mathbf{f}}(U) = {\det z_j^{f_i +N -i}\over \prod_{i 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 : (q_1,\ldots,q_N)\cdot (r_1,\ldots,r_N) = \sum r_i^*q_i on
HN, where
q* denotes the quaternion conjugate to
q. Realizing quaternions as 2 x 2 complex matrices, the group Sp(
N) is just the group of
block matrices (
qij) in SU(2
N) with :q_{ij}=\begin{pmatrix} \alpha_{ij}&\beta_{ij}\\ -\overline{\beta}_{ij}&\overline{\alpha}_{ij} \end{pmatrix}, where
αij and
βij are
complex numbers. Each matrix
U in Sp(
N) is conjugate to a block
diagonal matrix with entries :q_i=\begin{pmatrix} z_i&0\\ 0&\overline{z}_i \end{pmatrix}, where |
zi| = 1. Thus the eigenvalues of
U are (
zi±1). The irreducible representations of Sp(
N) are labelled by signatures :\mathbf{f} \,\colon \,f_1\ge f_2\ge \cdots \ge f_N\ge 0 where the
fi are integers. The character of the corresponding irreducible representation
σf is given by : \operatorname{Tr} \sigma_{\mathbf{f}}(U) = {\det z_j^{f_i +N -i +1 } - z_j^{-f_i - N +i -1}\over \prod (z_i-z_i^{-1})\cdot \prod_{i The branching rule from Sp(
N) to Sp(
N – 1) states that : Here
fN + 1 = 0 and the
multiplicity m(
f,
g) is given by : m(\mathbf{f},\mathbf{g})=\prod_{i=1}^N (a_i - b_i +1) where : a_1\ge b_1 \ge a_2 \ge b_2 \ge \cdots \ge a_N \ge b_N=0 is the non-increasing rearrangement of the 2
N non-negative integers (
fi), (
gj) and 0.
Example. The branching from U(2
N) to Sp(
N) relies on two identities of
Littlewood: : \begin{align} & \sum_{f_1\ge f_2\ge f_N\ge 0} \operatorname{Tr}\Pi_{\mathbf{f},0}(z_1,z_1^{-1},\ldots, z_N,z_N^{-1}) \cdot \operatorname{Tr}\pi_{\mathbf{f}}(t_1,\ldots,t_N) \\[5pt] = {} & \sum_{f_1\ge f_2\ge f_N\ge 0} \operatorname{Tr}\sigma_{\mathbf{f}}(z_1,\ldots, z_N) \cdot \operatorname{Tr}\pi_{\mathbf{f}}(t_1,\ldots,t_N)\cdot \prod_{i where Π
f,0 is the irreducible representation of
U(2
N) with signature
f1 ≥ ··· ≥
fN ≥ 0 ≥ ··· ≥ 0. :\prod_{i where
fi ≥ 0. The branching rule from U(2
N) 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
k of
U(
N) in the tensor product
g \otimes
h. It is given combinatorially by the Littlewood–Richardson rule, the number of lattice permutations of the
skew diagram k/
h of weight
g. • f_1\ge f_2 \ge \cdots \ge f_{n-1}\ge|f_n| for
N = 2
n; • f_1 \ge f_2 \ge \cdots \ge f_n \ge 0 for
N = 2
n+1. The
fi are taken in
Z for ordinary representations and in ½ +
Z for spin representations. In fact if an orthogonal matrix
U has eigenvalues
zi±1 for 1 ≤
i ≤
n, then the character of the corresponding irreducible representation
f is given by : \operatorname{Tr} \, \pi_{\mathbf{f}}(U) = {\det (z_j^{f_i +n -i} + z_j^{-f_i-n +i}) \over \prod_{i for
N = 2
n and by :\operatorname{Tr} \pi_{\mathbf{f}}(U) = {\det (z_j^{f_i +1/2 +n -i} - z_j^{-f_i -1/2-n +i})\over \prod_{i for
N = 2
n+1. The branching rules from SO(
N) to SO(
N – 1) state that : \pi_{\mathbf{g}} for
N = 2
n + 1 and : \pi_{\mathbf{g}} for
N = 2
n, where the differences
fi −
gi must be integers. ==Gelfand–Tsetlin basis== Since the branching rules from U(N) to U(N-1) or SO(N) to SO(N-1) have multiplicity one, the irreducible summands corresponding to smaller and smaller
N will eventually terminate in one-dimensional subspaces. In this way
Gelfand and
Tsetlin were able to obtain a basis of any irreducible representation of U(N) or SO(N) labelled by a chain of interleaved signatures, called a
Gelfand–Tsetlin pattern. Explicit formulas for the action of the Lie algebra on the
Gelfand–Tsetlin basis are given in . Specifically, for N=3, the Gelfand-Testlin basis of the irreducible representation of SO(3) with dimension 2l+1 is given by the complex
spherical harmonics \{Y_m^l | -l\leq m\leq l\}. For the remaining classical group Sp(N), the branching is no longer multiplicity free, so that if
V and
W are irreducible representation of Sp(N-1) and Sp(N) the space of intertwiners Hom_{Sp(N-1)}(V,W) can have dimension greater than one. It turns out that the
Yangian Y(\mathfrak{gl}_2), a
Hopf algebra introduced by
Ludwig Faddeev and
collaborators, acts irreducibly on this multiplicity space, a fact which enabled to extend the construction of Gelfand–Tsetlin bases to Sp(N). == Clifford's theorem ==