Restricted representation

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

${\displaystyle \rho {|}_H}$

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

${\displaystyle \rho {|}_H(h)=\rho (h).}$

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

${\displaystyle \mathbf{f}:f_1\ge f_2\ge \cdots \ge f_N}$

where the f_i are integers. In fact if a unitary matrix U has eigenvalues z_i, then the character of the corresponding irreducible representation π_f is given by

${\displaystyle \mathrm{Tr}{\pi }_{\mathbf{f}}(U)=\frac{det{z}_j^{f_i+N-i}}{\prod _{i<j}(z_i-z_j)}.}$

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 H^N which commute with right multiplication by the quaternions H and preserve the H-valued hermitian inner product

${\displaystyle (q_1,\dots ,q_N)\cdot (r_1,\dots ,r_N)=\sum r_i^{\ast }{q}_i}$

on H^N, 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 (q_ij) in SU(2N) with

${\displaystyle q_{ij}=\left(\begin{array}{cc}{\alpha }_{ij}& {\beta }_{ij}\\ -{\overline{\beta }}_{ij}& {\overline{\alpha }}_{ij}\end{array}\right),}$

where α_ij and β_ij are complex numbers.

Each matrix U in Sp(N) is conjugate to a block diagonal matrix with entries

${\displaystyle q_i=\left(\begin{array}{cc}{z}_i& 0\\ 0& {\overline{z}}_i\end{array}\right),}$

where |z_i| = 1. Thus the eigenvalues of U are (z_i^±1). The irreducible representations of Sp(N) are labelled by signatures

${\displaystyle \mathbf{f}:f_1\ge f_2\ge \cdots \ge f_N\ge 0}$

where the f_i are integers. The character of the corresponding irreducible representation σ_f is given by

${\displaystyle \mathrm{Tr}{\sigma }_{\mathbf{f}}(U)=\frac{det{z}_j^{f_i+N-i+1}-z_j^{-f_i-N+i-1}}{\prod (z_i-z_i^{-1})\cdot \prod _{i<j}(z_i+z_i^{-1}-z_j-z_j^{-1})}.}$

The branching rule from Sp(N) to Sp(N – 1) states that

Here f_{N + 1} = 0 and the multiplicity m(f, g) is given by

${\displaystyle m(\mathbf{f},\mathbf{g})=\prod _{i=1}^N(a_i-b_i+1)}$

where

${\displaystyle a_1\ge b_1\ge a_2\ge b_2\ge \cdots \ge a_N\ge b_N=0}$

is the non-increasing rearrangement of the 2N non-negative integers (f_i), (g_j) and 0.

Example. The branching from U(2N) to Sp(N) relies on two identities of Littlewood:

${\displaystyle \begin{array}{cc}& \sum _{f_1\ge f_2\ge f_N\ge 0}\mathrm{Tr}{\mathrm{\Pi }}_{\mathbf{f},0}(z_1,z_1^{-1},\dots ,z_N,z_N^{-1})\cdot \mathrm{Tr}{\pi }_{\mathbf{f}}(t_1,\dots ,t_N)\\ =& \sum _{f_1\ge f_2\ge f_N\ge 0}\mathrm{Tr}{\sigma }_{\mathbf{f}}(z_1,\dots ,z_N)\cdot \mathrm{Tr}{\pi }_{\mathbf{f}}(t_1,\dots ,t_N)\cdot \prod _{i<j}(1-z_i{z}_j{)}^{-1},\end{array}}$

where Π_f,0 is the irreducible representation of U(2N) with signature f₁ ≥ ··· ≥ f_N ≥ 0 ≥ ··· ≥ 0.

${\displaystyle \prod _{i<j}(1-z_i{z}_j{)}^{-1}=\sum _{f_{2i-1}=f_{2i}}\mathrm{Tr}{\pi }_f(z_1,\dots ,z_N),}$

where f_i ≥ 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 π_k of U(N) in the tensor product π_g ${\displaystyle \otimes }$ π_h. It is given combinatorially by the Littlewood–Richardson rule, the number of lattice permutations of the skew diagram k/h of weight g.

There is an extension of Littlewood's branching rule to arbitrary signatures due to Sundaram (1990, p. 203). The Littlewood–Richardson coefficients M (g, h; f) are extended to allow the signature f to have 2N parts but restricting g to have even column-lengths (g_{2i – 1} = g_2i). In this case the formula reads

where M_N (g, h; f) counts the number of lattice permutations of f/h of weight g are counted for which 2j + 1 appears no lower than row N + j of f for 1 ≤ j ≤ |g|/2.

Example. The special orthogonal group SO(N) has irreducible ordinary and spin representations labelled by signatures

• ${\displaystyle f_1\ge f_2\ge \cdots \ge f_{n-1}\ge |f_n|}$ for N = 2n;
• ${\displaystyle f_1\ge f_2\ge \cdots \ge f_n\ge 0}$ for N = 2n+1.

The f_i are taken in Z for ordinary representations and in ½ + Z for spin representations. In fact if an orthogonal matrix U has eigenvalues z_i^±1 for 1 ≤ i ≤ n, then the character of the corresponding irreducible representation π_f is given by

${\displaystyle \mathrm{Tr}{\pi }_{\mathbf{f}}(U)=\frac{det(z_j^{f_i+n-i}+z_j^{-f_i-n+i})}{\prod _{i<j}(z_i+z_i^{-1}-z_j-z_j^{-1})}}$

for N = 2n and by

${\displaystyle \mathrm{Tr}{\pi }_{\mathbf{f}}(U)=\frac{det(z_j^{f_i+1/2+n-i}-z_j^{-f_i-1/2-n+i})}{\prod _{i<j}(z_i+z_i^{-1}-z_j-z_j^{-1})\cdot \prod _k(z_k^{1/2}-z_k^{-1/2})}}$

for N = 2n+1.

The branching rules from SO(N) to SO(N – 1) state that

for N = 2n + 1 and

for N = 2n, where the differences f_i − g_i must be integers.