Editing Restricted representation

In [[group theory]], '''restriction''' forms a [[Group representation|representation]] of a [[subgroup]] using a known representation of the whole [[group (mathematics)|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 [[multiplet]]s, as in the [[Stark effect|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.<ref>{{harvnb|Weyl|1946|pp=159–160}}.</ref> Restriction can be generalized to other [[group homomorphism]]s and to other [[ring (mathematics)|rings]].

For any group ''G'', its [[subgroup]] ''H'', and a [[linear representation]] ''ρ'' of ''G'', the restriction of ''ρ'' to ''H'', denoted

: <math> \rho \, \Big|_H </math> 

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

: <math> \rho\,\Big|_H(h) = \rho(h). </math>

== 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.

==Gelfand–Tsetlin basis==<!-- Linked from [[Gelfand–Tsetlin integrable system]] -->
{{see also|Gelfand–Tsetlin integrable system}}
Since the branching rules from <math>U(N)</math> to <math>U(N-1)</matH> or <math>SO(N)</math> to <matH>SO(N-1)</math> have multiplicity one, the irreducible summands corresponding to smaller and smaller ''N'' will eventually terminate in one-dimensional subspaces. In this way [[I. M. Gelfand|Gelfand]] and [[Michael Tsetlin|Tsetlin]] were able to obtain a basis of any irreducible representation of <math>U(N)</math> or <math>SO(N)</math> 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 {{harvtxt|Želobenko|1973}}. Specifically, for <math>N=3</math>, the Gelfand-Testlin basis of the irreducible representation of <math>SO(3)</math> with dimension <math>2l+1</math> is given by the complex [[spherical harmonics]] <math>\{Y_m^l | -l\leq m\leq l\}</math>.

For the remaining classical group <math>Sp(N)</math>, the branching is no longer multiplicity free, so that if ''V'' and ''W'' are irreducible representation of <math>Sp(N-1)</math> and <math>Sp(N)</math> the space of intertwiners <math>Hom_{Sp(N-1)}(V,W)</math> can have dimension greater than one. It turns out that the [[Yangian]] <math>Y(\mathfrak{gl}_2)</math>, a [[Hopf algebra]] introduced by [[Ludwig Faddeev]] and [[LOMI|collaborators]], acts irreducibly on this  multiplicity space, a fact which enabled {{harvtxt|Molev|2006}} to extend the construction of Gelfand–Tsetlin bases to <math>Sp(N)</math>.<ref>G. I. Olshanski had shown that the twisted Yangian <math>Y^-(\mathfrak{gl}_2)</math>, a sub-Hopf algebra of <math>Y(\mathfrak{gl}_2)</math>, acts naturally on the space of intertwiners. Its natural irreducible representations correspond to tensor products of the composition of point evaluations with irreducible representations of <math>\mathfrak{gl}</math><sub>2</sub>. These extend to the Yangian <math>Y(\mathfrak{gl})</math> and give a representation theoretic explanation of the product form of the branching coefficients.</ref>

== Clifford's theorem ==
{{main|Clifford theory}}
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 of a subgroup|index]]:<ref>{{harvnb|Weyl|1946|pp=159–160, 311}}</ref>

'''Theorem'''. Let {{pi}}: ''G'' <math>\rightarrow </math> GL(''n'',''K'') be an irreducible representation with ''K'' a [[field (mathematics)|field]]. Then 
the restriction of {{pi}} 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&nbsp;''G''.

Twenty years later [[George Mackey]] found a more precise version of this result for the restriction of irreducible [[unitary representation]]s of [[locally compact group]]s to closed normal subgroups in what has become known as the "Mackey machine" or "Mackey normal subgroup analysis".<ref>{{citation|first=George W.|last=Mackey|authorlink=George Mackey|title=The theory of unitary group representations|series=Chicago Lectures in Mathematics|year=1976|isbn=978-0-226-50052-2}}</ref>

== Abstract algebraic setting ==
{{Main|Frobenius reciprocity}}
From the point of view of [[category theory]], restriction is an instance of a [[forgetful functor]]. This functor is [[exact functor|exact]], and its [[adjoint functor|left adjoint functor]] is called ''induction''. 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. This is especially true whenever the representations have the property of [[completely reducible|complete reducibility]], for example, in [[representation theory of finite groups]] over a [[field (algebra)|field]] of [[characteristic zero]].

== Generalizations ==
This rather evident construction may be extended in numerous and significant ways. For instance we may take any group homomorphism φ from ''H'' to ''G'', instead of the [[inclusion map]], and define the restricted representation of ''H'' by the composition

: <math> \rho\circ\varphi \, </math>

We may also apply the idea to other categories in [[abstract algebra]]: [[associative algebra]]s, rings, [[Lie algebra]]s, [[Lie superalgebra]]s, Hopf algebras to name some. Representations or [[module (mathematics)|module]]s ''restrict'' to subobjects, or via homomorphisms.

==Notes==
{{reflist|2}}

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[[Category:Representation theory]]
[[Category:Algebraic combinatorics]]