Editing Restricted representation (section)

== Classical branching rules ==
'''Classical branching rules''' describe the restriction of an irreducible complex representation ({{pi}},&nbsp;''V'') of a [[classical group]] ''G'' to a classical subgroup ''H'', i.e. the multiplicity with which an irreducible representation (''σ'',&nbsp;''W'') of ''H'' occurs in&nbsp;{{pi}}. By Frobenius reciprocity for [[compact group]]s, this is equivalent to finding the multiplicity of {{pi}} in the [[induced representations|unitary representation induced]] from σ. Branching rules for the classical groups were determined by
* {{harvtxt|Weyl|1946}} between successive [[unitary group]]s;
* {{harvtxt|Murnaghan|1938}} between successive [[special orthogonal group]]s and [[unitary symplectic group]]s;
* {{harvtxt|Littlewood|1950}} from the unitary groups to the unitary symplectic groups and special orthogonal groups.

The results are usually expressed graphically using [[Young diagram]]s to encode the signatures used classically to label irreducible representations, familiar from [[invariant theory|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]].<ref name="Weyl">{{harvnb|Weyl|1946}}</ref><ref>{{harvnb|Želobenko|1973}}</ref> A systematic modern interpretation has been given by {{harvtxt|Howe|1995}} in the context of his theory of [[Reductive dual pair|dual pair]]s. The special case where σ is the trivial representation of ''H'' was first used extensively by [[Hua Loo-keng|Hua]] in his work on the [[reproducing kernel|Szegő kernels]] of [[hermitian symmetric space|bounded symmetric domain]]s in [[several complex variables]], where the [[Shilov boundary]] has the form ''G''/''H''.<ref>{{harvnb|Helgason|1978}}</ref><ref>{{harvnb|Hua|1963}}</ref> More generally the [[Zonal spherical function#Cartan–Helgason theorem|Cartan-Helgason theorem]] gives the decomposition when ''G''/''H'' is a compact symmetric space, in which case all multiplicities are one;<ref>{{harvnb|Helgason|1984|pp=534–543}}</ref> a generalization to arbitrary σ has since been obtained by {{harvtxt|Kostant|2004}}. Similar geometric considerations have also been used by {{harvtxt|Knapp|2003}} to rederive Littlewood's rules, which involve the celebrated [[Littlewood–Richardson rule]]s for tensoring irreducible representations of the unitary groups.
{{harvtxt|Littelmann|1995}} has found generalizations of these rules to arbitrary compact semisimple [[Lie group]]s, using his [[Littelmann path model|path model]], an approach to representation theory close in spirit to the theory of [[crystal basis|crystal bases]] of [[George Lusztig|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]].<ref name="Goodman">{{harvnb|Goodman|Wallach|1998}}</ref><ref name="Macdonald">{{harvnb|Macdonald|1979}}</ref>

'''Example'''. The unitary group ''U''(''N'') has irreducible representations labelled by signatures

:<math>\mathbf{f} \,\colon \,f_1\ge f_2\ge \cdots \ge f_N</math>

where the ''f''<sub>''i''</sub> are integers. In fact if a unitary matrix ''U'' has eigenvalues ''z''<sub>''i''</sub>, then the character of the corresponding irreducible representation {{pi}}<sub>'''f'''</sub> is given by

:<math> \operatorname{Tr} \pi_{\mathbf{f}}(U) = {\det z_j^{f_i +N -i}\over \prod_{i<j} (z_i-z_j)}.</math>

The branching rule from ''U''(''N'') to ''U''(''N''&nbsp;–&nbsp;1) states that

:{| border="1" cellspacing="0" cellpadding="5"
|<math>\pi_{\mathbf{f}}|_{U(N-1)}= \bigoplus_{f_1\ge g_1 \ge f_2\ge g_2\ge \cdots \ge f_{N-1}\ge g_{N-1}\ge f_N} \pi_{\mathbf{g}}</math>
|}

'''Example'''. The unitary symplectic group or [[quaternionic unitary group]], denoted Sp(''N'') or  ''U''(''N'', '''H'''), is the group of all transformations of
'''H'''<sup>''N''</sup> which commute with right multiplication by the [[quaternions]] '''H''' and preserve the '''H'''-valued hermitian inner product

:<math> (q_1,\ldots,q_N)\cdot (r_1,\ldots,r_N) = \sum r_i^*q_i</math>

on '''H'''<sup>''N''</sup>, 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 matrix|block matrices]] (''q''<sub>''ij''</sub>) in SU(2''N'') with

:<math>q_{ij}=\begin{pmatrix}
\alpha_{ij}&\beta_{ij}\\
-\overline{\beta}_{ij}&\overline{\alpha}_{ij}
\end{pmatrix},</math>

where ''α''<sub>''ij''</sub> and ''β''<sub>''ij''</sub> are [[complex number]]s.

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

:<math>q_i=\begin{pmatrix}
z_i&0\\
0&\overline{z}_i
\end{pmatrix},</math>

where |''z''<sub>''i''</sub>| = 1. Thus the eigenvalues of ''U'' are (''z''<sub>''i''</sub><sup>±1</sup>). The irreducible representations of Sp(''N'') are labelled by signatures

:<math>\mathbf{f} \,\colon \,f_1\ge f_2\ge \cdots \ge f_N\ge 0</math>

where the ''f''<sub>''i''</sub> are integers. The character of the corresponding irreducible representation ''σ''<sub>'''f'''</sub> is given by<ref>{{harvnb|Weyl|1946|p=218}}</ref>

: <math> \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<j} (z_i +z_i^{-1} - z_j - z_j^{-1})}.</math>

The branching rule from Sp(''N'') to Sp(''N''&nbsp;–&nbsp;1) states that<ref>{{harvnb|Goodman|Wallach|1998|pp=351–352,365–370}}</ref>

:{| border="1" cellspacing="0" cellpadding="5"
|<math>\sigma_{\mathbf{f}}|_{\mathrm{Sp}(N-1)}= \bigoplus_{f_i \ge g_i\ge f_{i+2}} m(\mathbf{f},\mathbf{g}) \sigma_{\mathbf{g}}</math>
|}

Here ''f''<sub>''N'' + 1</sub> = 0 and the [[multiplicity (mathematics)|multiplicity]] ''m''('''f''', '''g''') is given by

:<math> m(\mathbf{f},\mathbf{g})=\prod_{i=1}^N (a_i - b_i +1)</math>

where

:<math> a_1\ge b_1 \ge a_2 \ge b_2 \ge \cdots \ge a_N \ge b_N=0</math>

is the non-increasing rearrangement of the 2''N'' non-negative integers (''f''<sub>i</sub>), (''g''<sub>''j''</sub>) and 0.

'''Example'''. The branching from U(2''N'') to Sp(''N'') relies on two identities of [[Dudley E. Littlewood|Littlewood]]:<ref>{{harvnb|Littlewood|1950}}</ref><ref>{{harvnb|Weyl|1946|pp=216–222}}</ref><ref>{{harvnb|Koike|Terada|1987}}</ref><ref>{{harvnb|Macdonald|1979|p=46}}</ref>

:<math>
\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<j} (1-z_iz_j)^{-1},
\end{align}
</math>

where Π<sub>'''f''',0</sub> is the irreducible representation of ''U''(2''N'') with signature ''f''<sub>1</sub> ≥ ··· ≥ ''f''<sub>''N''</sub> ≥ 0 ≥ ··· ≥ 0.

:<math>\prod_{i<j} (1-z_iz_j)^{-1} =  \sum_{f_{2i-1}=f_{2i}} \operatorname{Tr} \pi_{f}(z_1,\ldots,z_N),</math>

where ''f''<sub>''i''</sub> ≥ 0.

The branching rule from U(2''N'') to Sp(''N'') is given by

:{| border="1" cellspacing="0" cellpadding="5"
|<math>\Pi_{\mathbf{f},0}|_{\mathrm{Sp}(N)}= \bigoplus_{\mathbf{h}, \,\,\mathbf{g},\,\, g_{2i-1}=g_{2i}} M(\mathbf{g}, \mathbf{h};\mathbf{f}) \sigma_{\mathbf{h}}</math>
|}

where all the signature are non-negative and the coefficient ''M'' ('''g''', '''h'''; '''k''') is the multiplicity of the irreducible representation {{pi}}<sub>'''k'''</sub> of ''U''(''N'') in the tensor product {{pi}}<sub>'''g'''</sub> <math>\otimes</math> {{pi}}<sub>'''h'''</sub>. It is given combinatorially by the Littlewood–Richardson rule, the number of lattice permutations of the [[Skew tableau#Skew tableaux|skew diagram]] '''k'''/'''h''' of weight '''g'''.<ref name="Macdonald" />

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

:{| border="1" cellspacing="0" cellpadding="5"
|<math>\Pi_{\mathbf{f}}|_{\operatorname{Sp}(N)}= \bigoplus_{\mathbf{h}, \,\,\mathbf{g},\,\, g_{2i-1}=g_{2i}} M_N(\mathbf{g}, \mathbf{h};\mathbf{f}) \sigma_{\mathbf{h}}</math>
|}

where ''M''<sub>''N''</sub> ('''g''', '''h'''; '''f''') counts the number of lattice permutations of '''f'''/'''h''' of weight '''g''' are counted for which 2''j'' + 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 representation]]s labelled by signatures<ref name="Weyl" /><ref name="Goodman" /><ref>{{harvnb|Littlewood|1950|pp=223–263}}</ref><ref>{{harvnb|Murnaghan|1938}}</ref>

* <math> f_1\ge f_2 \ge \cdots \ge f_{n-1}\ge|f_n|</math> for ''N'' = 2''n'';
*<math> f_1 \ge f_2 \ge \cdots \ge f_n \ge 0</math> for ''N'' = 2''n''+1.

The ''f''<sub>''i''</sub> are taken in '''Z''' for ordinary representations and in ½ + '''Z''' for spin representations. In fact if an orthogonal matrix ''U'' has eigenvalues ''z''<sub>''i''</sub><sup>±1</sup> for 1 ≤ ''i'' ≤ ''n'', then the character of the corresponding irreducible representation {{pi}}<sub>'''f'''</sub> is given by

: <math> \operatorname{Tr} \, \pi_{\mathbf{f}}(U) = {\det (z_j^{f_i +n -i} + z_j^{-f_i-n +i}) \over \prod_{i<j} (z_i +z_i^{-1}-z_j-z_j^{-1})}</math>

for ''N'' = 2''n'' and by

:<math>\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<j} (z_i +z_i^{-1}-z_j-z_j^{-1}) \cdot\prod_k(z_k^{1/2} -z_k^{-1/2})}</math>

for ''N'' = 2''n''+1.

The branching rules from SO(''N'') to  SO(''N''&nbsp;–&nbsp;1) state that<ref>{{harvnb|Goodman|Wallach|1998|p=351}}</ref>

:{| border="1" cellspacing="0" cellpadding="5"
|<math>\pi_{\mathbf{f}}|_{SO(2n)}= \bigoplus_{f_1\ge g_1 \ge f_2\ge g_2\ge \cdots \ge f_{n-1}\ge g_{n-1}\ge f_n \ge |g_n|} \pi_{\mathbf{g}}</math>
|}

for ''N'' = 2''n''&nbsp;+&nbsp;1 and

:{| border="1" cellspacing="0" cellpadding="5"
|<math>\pi_{\mathbf{f}}|_{SO(2n-1)}= \bigoplus_{f_1\ge g_1 \ge f_2\ge g_2\ge \cdots \ge f_{n-1}\ge g_{n-1}\ge |f_n|} \pi_{\mathbf{g}}</math>
|}

for ''N'' = 2''n'', where the differences ''f''<sub>''i''</sub>&nbsp;−&nbsp;''g''<sub>''i''</sub> must be integers.