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<h2 id="classical-branching-rules">Classical branching rules</h2> <p>Classical branching rules describe the restriction of an irreducible complex representation (π, <i>V</i>) of a <a href="/facts/Classical_group/TEV7cSvJ">classical group</a> <i>G</i> to a classical subgroup <i>H</i>, i.e. the multiplicity with which an irreducible representation (<i>σ</i>, <i>W</i>) of <i>H</i> occurs in π. By Frobenius reciprocity for <a href="/facts/Compact_group/Itl2HJaR">compact groups</a>, this is equivalent to finding the multiplicity of π in the <a href="/facts/Induced_representations/Jh25PGDP">unitary representation induced</a> from σ. Branching rules for the classical groups were determined by </p> <ul><li>Weyl (1946) between successive <a href="/facts/Unitary_group/lEvudIhU">unitary groups</a>;</li> <li>Murnaghan (1938) between successive <a href="/facts/Special_orthogonal_group/jKWM1KDM">special orthogonal groups</a> and <a href="/facts/Unitary_symplectic_group/G4CKBc7z">unitary symplectic groups</a>;</li> <li>Littlewood (1950) from the unitary groups to the unitary symplectic groups and special orthogonal groups.</li></ul> <p>The results are usually expressed graphically using <a href="/facts/Young_diagram/e5lI4bck">Young diagrams</a> to encode the signatures used classically to label irreducible representations, familiar from <a href="/facts/Invariant_theory/5fIUh9h3">classical invariant theory</a>. <a href="/facts/Hermann_Weyl/4C5HhyGU">Hermann Weyl</a> and <a href="/facts/Richard_Brauer/YlzG0z1w">Richard Brauer</a> discovered a systematic method for determining the branching rule when the groups <i>G</i> and <i>H</i> share a common <a href="/facts/Maximal_torus/f82Qln74">maximal torus</a>: in this case the <a href="/facts/Weyl_group/N9QG4Wyn">Weyl group</a> of <i>H</i> is a subgroup of that of <i>G</i>, so that the rule can be deduced from the <a href="/facts/Weyl_character_formula/nbbNv8oN">Weyl character formula</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> A systematic modern interpretation has been given by Howe (1995) in the context of his theory of <a href="/facts/Reductive_dual_pair/pKpCTJHT">dual pairs</a>. The special case where σ is the trivial representation of <i>H</i> was first used extensively by <a href="/facts/Hua_Loo-keng/g6TDG442">Hua</a> in his work on the <a href="/facts/Reproducing_kernel/VegkYAXb">Szegő kernels</a> of <a href="/facts/Hermitian_symmetric_space/fLBcoPbU">bounded symmetric domains</a> in <a href="/facts/Several_complex_variables/axwNpV4a">several complex variables</a>, where the <a href="/facts/Shilov_boundary/423ceqlR">Shilov boundary</a> has the form <i>G</i>/<i>H</i>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> More generally the <a href="/facts/Zonal_spherical_function/myzsFsrU">Cartan-Helgason theorem</a> gives the decomposition when <i>G</i>/<i>H</i> is a compact symmetric space, in which case all multiplicities are one;<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 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 <a href="/facts/Littlewood%E2%80%93Richardson_rule/8NSx5K36">Littlewood–Richardson rules</a> for tensoring irreducible representations of the unitary groups. Littelmann (1995) has found generalizations of these rules to arbitrary compact semisimple <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>, using his <a href="/facts/Littelmann_path_model/DFuwMww1">path model</a>, an approach to representation theory close in spirit to the theory of <a href="/facts/Crystal_basis/sb1qXgwa">crystal bases</a> of <a href="/facts/George_Lusztig/htWIjRNh">Lusztig</a> and <a href="/facts/Kashiwara/SgJyEX48">Kashiwara</a>. 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, <a href="/facts/Algebraic_combinatorics/9b47n3Fh">algebraic combinatorics</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p><p>Example. The unitary group <i>U</i>(<i>N</i>) has irreducible representations labelled by signatures </p> f : f 1 ≥ f 2 ≥ ⋯ ≥ f N {\displaystyle \mathbf {f} \,\colon \,f_{1}\geq f_{2}\geq \cdots \geq f_{N}} <p>where the <i>f</i><i>i</i> are integers. In fact if a unitary matrix <i>U</i> has eigenvalues <i>z</i><i>i</i>, then the character of the corresponding irreducible representation πf is given by </p> Tr π f ( U ) = det z j f i + N − i ∏ i < j ( z i − z j ) . {\displaystyle \operatorname {Tr} \pi _{\mathbf {f} }(U)={\det z_{j}^{f_{i}+N-i} \over \prod _{i<j}(z_{i}-z_{j})}.} <p>The branching rule from <i>U</i>(<i>N</i>) to <i>U</i>(<i>N</i> – 1) states that </p> <table><tbody><tr><td> π f | U ( N − 1 ) = ⨁ f 1 ≥ g 1 ≥ f 2 ≥ g 2 ≥ ⋯ ≥ f N − 1 ≥ g N − 1 ≥ f N π g {\displaystyle \pi _{\mathbf {f} }|_{U(N-1)}=\bigoplus _{f_{1}\geq g_{1}\geq f_{2}\geq g_{2}\geq \cdots \geq f_{N-1}\geq g_{N-1}\geq f_{N}}\pi _{\mathbf {g} }} </td></tr></tbody></table> <p>Example. The unitary symplectic group or <a href="/facts/Quaternionic_unitary_group/G4CKBc7z">quaternionic unitary group</a>, denoted Sp(<i>N</i>) or <i>U</i>(<i>N</i>, H), is the group of all transformations of H<i>N</i> which commute with right multiplication by the <a href="/facts/Quaternions/T3tnu7mX">quaternions</a> H and preserve the H-valued hermitian inner product </p> ( q 1 , … , q N ) ⋅ ( r 1 , … , r N ) = ∑ r i ∗ q i {\displaystyle (q_{1},\ldots ,q_{N})\cdot (r_{1},\ldots ,r_{N})=\sum r_{i}^{*}q_{i}} <p>on H<i>N</i>, where <i>q</i>* denotes the quaternion conjugate to <i>q</i>. Realizing quaternions as 2 x 2 complex matrices, the group Sp(<i>N</i>) is just the group of <a href="/facts/Block_matrix/fPI8HX9f">block matrices</a> (<i>q</i><i>ij</i>) in SU(2<i>N</i>) with </p> q i j = ( α i j β i j − β ¯ i j α ¯ i j ) , {\displaystyle q_{ij}={\begin{pmatrix}\alpha _{ij}&\beta _{ij}\\-{\overline {\beta }}_{ij}&{\overline {\alpha }}_{ij}\end{pmatrix}},} <p>where <i>α</i><i>ij</i> and <i>β</i><i>ij</i> are <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>. </p><p>Each matrix <i>U</i> in Sp(<i>N</i>) is conjugate to a block diagonal matrix with entries </p> q i = ( z i 0 0 z ¯ i ) , {\displaystyle q_{i}={\begin{pmatrix}z_{i}&0\\0&{\overline {z}}_{i}\end{pmatrix}},} <p>where |<i>z</i><i>i</i>| = 1. Thus the eigenvalues of <i>U</i> are (<i>z</i><i>i</i>±1). The irreducible representations of Sp(<i>N</i>) are labelled by signatures </p> f : f 1 ≥ f 2 ≥ ⋯ ≥ f N ≥ 0 {\displaystyle \mathbf {f} \,\colon \,f_{1}\geq f_{2}\geq \cdots \geq f_{N}\geq 0} <p>where the <i>f</i><i>i</i> are integers. The character of the corresponding irreducible representation <i>σ</i>f is given by<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> Tr σ f ( U ) = det z j f i + N − i + 1 − z j − f i − N + i − 1 ∏ ( z i − z i − 1 ) ⋅ ∏ i < j ( z i + z i − 1 − z j − z j − 1 ) . {\displaystyle \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})}.} <p>The branching rule from Sp(<i>N</i>) to Sp(<i>N</i> – 1) states that<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <table><tbody><tr><td> σ f | S p ( N − 1 ) = ⨁ f i ≥ g i ≥ f i + 2 m ( f , g ) σ g {\displaystyle \sigma _{\mathbf {f} }|_{\mathrm {Sp} (N-1)}=\bigoplus _{f_{i}\geq g_{i}\geq f_{i+2}}m(\mathbf {f} ,\mathbf {g} )\sigma _{\mathbf {g} }} </td></tr></tbody></table> <p>Here <i>f</i><i>N</i> + 1 = 0 and the <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a> <i>m</i>(f, g) is given by </p> m ( f , g ) = ∏ i = 1 N ( a i − b i + 1 ) {\displaystyle m(\mathbf {f} ,\mathbf {g} )=\prod _{i=1}^{N}(a_{i}-b_{i}+1)} <p>where </p> a 1 ≥ b 1 ≥ a 2 ≥ b 2 ≥ ⋯ ≥ a N ≥ b N = 0 {\displaystyle a_{1}\geq b_{1}\geq a_{2}\geq b_{2}\geq \cdots \geq a_{N}\geq b_{N}=0} <p>is the non-increasing rearrangement of the 2<i>N</i> non-negative integers (<i>f</i>i), (<i>g</i><i>j</i>) and 0. </p><p>Example. The branching from U(2<i>N</i>) to Sp(<i>N</i>) relies on two identities of <a href="/facts/Dudley_E._Littlewood/Q5Y5d7g1">Littlewood</a>:<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> ∑ f 1 ≥ f 2 ≥ f N ≥ 0 Tr Π f , 0 ( z 1 , z 1 − 1 , … , z N , z N − 1 ) ⋅ Tr π f ( t 1 , … , t N ) = ∑ f 1 ≥ f 2 ≥ f N ≥ 0 Tr σ f ( z 1 , … , z N ) ⋅ Tr π f ( t 1 , … , t N ) ⋅ ∏ i < j ( 1 − z i z j ) − 1 , {\displaystyle {\begin{aligned}&\sum _{f_{1}\geq f_{2}\geq f_{N}\geq 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}\geq f_{2}\geq f_{N}\geq 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_{i}z_{j})^{-1},\end{aligned}}} <p>where Πf,0 is the irreducible representation of <i>U</i>(2<i>N</i>) with signature <i>f</i>1 ≥ ··· ≥ <i>f</i><i>N</i> ≥ 0 ≥ ··· ≥ 0. </p> ∏ i < j ( 1 − z i z j ) − 1 = ∑ f 2 i − 1 = f 2 i Tr π f ( z 1 , … , z N ) , {\displaystyle \prod _{i<j}(1-z_{i}z_{j})^{-1}=\sum _{f_{2i-1}=f_{2i}}\operatorname {Tr} \pi _{f}(z_{1},\ldots ,z_{N}),} <p>where <i>f</i><i>i</i> ≥ 0. </p><p>The branching rule from U(2<i>N</i>) to Sp(<i>N</i>) is given by </p> <table><tbody><tr><td> Π f , 0 | S p ( N ) = ⨁ h , g , g 2 i − 1 = g 2 i M ( g , h ; f ) σ h {\displaystyle \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} }} </td></tr></tbody></table> <p>where all the signature are non-negative and the coefficient <i>M</i> (g, h; k) is the multiplicity of the irreducible representation πk of <i>U</i>(<i>N</i>) in the tensor product πg ⊗ {\displaystyle \otimes } πh. It is given combinatorially by the Littlewood–Richardson rule, the number of lattice permutations of the <a href="/facts/Skew_tableau/e5lI4bck">skew diagram</a> k/h of weight g.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p>There is an extension of Littlewood's branching rule to arbitrary signatures due to Sundaram (1990, p. 203). The Littlewood–Richardson coefficients <i>M</i> (g, h; f) are extended to allow the signature f to have 2<i>N</i> parts but restricting g to have even column-lengths (<i>g</i>2<i>i</i> – 1 = <i>g</i>2<i>i</i>). In this case the formula reads </p> <table><tbody><tr><td> Π f | Sp ( N ) = ⨁ h , g , g 2 i − 1 = g 2 i M N ( g , h ; f ) σ h {\displaystyle \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} }} </td></tr></tbody></table> <p>where <i>M</i><i>N</i> (g, h; f) counts the number of lattice permutations of f/h of weight g are counted for which 2<i>j</i> + 1 appears no lower than row <i>N</i> + <i>j</i> of f for 1 ≤ <i>j</i> ≤ |<i>g</i>|/2. </p><p>Example. The special orthogonal group SO(<i>N</i>) has irreducible ordinary and <a href="/facts/Spin_representation/upLgk4zS">spin representations</a> labelled by signatures<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> <ul><li> f 1 ≥ f 2 ≥ ⋯ ≥ f n − 1 ≥ | f n | {\displaystyle f_{1}\geq f_{2}\geq \cdots \geq f_{n-1}\geq |f_{n}|} for <i>N</i> = 2<i>n</i>;</li> <li> f 1 ≥ f 2 ≥ ⋯ ≥ f n ≥ 0 {\displaystyle f_{1}\geq f_{2}\geq \cdots \geq f_{n}\geq 0} for <i>N</i> = 2<i>n</i>+1.</li></ul> <p>The <i>f</i><i>i</i> are taken in Z for ordinary representations and in ½ + Z for spin representations. In fact if an orthogonal matrix <i>U</i> has eigenvalues <i>z</i><i>i</i>±1 for 1 ≤ <i>i</i> ≤ <i>n</i>, then the character of the corresponding irreducible representation πf is given by </p> Tr π f ( U ) = det ( z j f i + n − i + z j − f i − n + i ) ∏ i < j ( z i + z i − 1 − z j − z j − 1 ) {\displaystyle \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})}} <p>for <i>N</i> = 2<i>n</i> and by </p> Tr π f ( U ) = det ( z j f i + 1 / 2 + n − i − z j − f i − 1 / 2 − n + i ) ∏ i < j ( z i + z i − 1 − z j − z j − 1 ) ⋅ ∏ k ( z k 1 / 2 − z k − 1 / 2 ) {\displaystyle \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})}} <p>for <i>N</i> = 2<i>n</i>+1. </p><p>The branching rules from SO(<i>N</i>) to SO(<i>N</i> – 1) state that<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> <table><tbody><tr><td> π f | S O ( 2 n ) = ⨁ f 1 ≥ g 1 ≥ f 2 ≥ g 2 ≥ ⋯ ≥ f n − 1 ≥ g n − 1 ≥ f n ≥ | g n | π g {\displaystyle \pi _{\mathbf {f} }|_{SO(2n)}=\bigoplus _{f_{1}\geq g_{1}\geq f_{2}\geq g_{2}\geq \cdots \geq f_{n-1}\geq g_{n-1}\geq f_{n}\geq |g_{n}|}\pi _{\mathbf {g} }} </td></tr></tbody></table> <p>for <i>N</i> = 2<i>n</i> + 1 and </p> <table><tbody><tr><td> π f | S O ( 2 n − 1 ) = ⨁ f 1 ≥ g 1 ≥ f 2 ≥ g 2 ≥ ⋯ ≥ f n − 1 ≥ g n − 1 ≥ | f n | π g {\displaystyle \pi _{\mathbf {f} }|_{SO(2n-1)}=\bigoplus _{f_{1}\geq g_{1}\geq f_{2}\geq g_{2}\geq \cdots \geq f_{n-1}\geq g_{n-1}\geq |f_{n}|}\pi _{\mathbf {g} }} </td></tr></tbody></table> <p>for <i>N</i> = 2<i>n</i>, where the differences <i>f</i><i>i</i> − <i>g</i><i>i</i> must be integers. </p> <h2 id="gelfandtsetlin-basis">Gelfand–Tsetlin basis</h2> <p class="note">See also: <a href="/facts/Gelfand%E2%80%93Tsetlin_integrable_system/wc0280J1">Gelfand–Tsetlin integrable system</a></p> <p>Since the branching rules from U ( N ) {\displaystyle U(N)} to U ( N − 1 ) {\displaystyle U(N-1)} or S O ( N ) {\displaystyle SO(N)} to S O ( N − 1 ) {\displaystyle SO(N-1)} have multiplicity one, the irreducible summands corresponding to smaller and smaller <i>N</i> will eventually terminate in one-dimensional subspaces. In this way <a href="/facts/I._M._Gelfand/TJeJouKs">Gelfand</a> and Tsetlin were able to obtain a basis of any irreducible representation of U ( N ) {\displaystyle U(N)} or S O ( N ) {\displaystyle 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 Želobenko (1973). Specifically, for N = 3 {\displaystyle N=3} , the Gelfand-Testlin basis of the irreducible representation of S O ( 3 ) {\displaystyle SO(3)} with dimension 2 l + 1 {\displaystyle 2l+1} is given by the complex <a href="/facts/Spherical_harmonics/bn98Gv2L">spherical harmonics</a> { Y m l | − l ≤ m ≤ l } {\displaystyle \{Y_{m}^{l}|-l\leq m\leq l\}} . </p><p>For the remaining classical group S p ( N ) {\displaystyle Sp(N)} , the branching is no longer multiplicity free, so that if <i>V</i> and <i>W</i> are irreducible representation of S p ( N − 1 ) {\displaystyle Sp(N-1)} and S p ( N ) {\displaystyle Sp(N)} the space of intertwiners H o m S p ( N − 1 ) ( V , W ) {\displaystyle Hom_{Sp(N-1)}(V,W)} can have dimension greater than one. It turns out that the <a href="/facts/Yangian/3EKUD0wQ">Yangian</a> Y ( g l 2 ) {\displaystyle Y({\mathfrak {gl}}_{2})} , a <a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebra</a> introduced by <a href="/facts/Ludwig_Faddeev/gR8t21SP">Ludwig Faddeev</a> and <a href="/facts/LOMI/F1CdJq8k">collaborators</a>, acts irreducibly on this multiplicity space, a fact which enabled Molev (2006) to extend the construction of Gelfand–Tsetlin bases to S p ( N ) {\displaystyle Sp(N)} .<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p> <h2 id="cliffords-theorem">Clifford's theorem</h2> <p class="note">Main article: <a href="/facts/Clifford_theory/32sQWx2j">Clifford theory</a></p> <p>In 1937 <a href="/facts/Alfred_H._Clifford/6jpc6VPA">Alfred H. Clifford</a> proved the following result on the restriction of finite-dimensional irreducible representations from a group <i>G</i> to a normal subgroup <i>N</i> of finite <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a>:<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> </p><p>Theorem. Let π: <i>G</i> → {\displaystyle \rightarrow } GL(<i>n</i>,<i>K</i>) be an irreducible representation with <i>K</i> a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. Then the restriction of π to <i>N</i> breaks up into a direct sum of irreducible representations of <i>N</i> of equal dimensions. These irreducible representations of <i>N</i> lie in one orbit for the action of <i>G</i> by conjugation on the equivalence classes of irreducible representations of <i>N</i>. In particular the number of distinct summands is no greater than the index of <i>N</i> in <i>G</i>. </p><p>Twenty years later <a href="/facts/George_Mackey/uoeY23Qv">George Mackey</a> found a more precise version of this result for the restriction of irreducible <a href="/facts/Unitary_representation/W1XRatG8">unitary representations</a> of <a href="/facts/Locally_compact_group/AGPC8vnY">locally compact groups</a> to closed normal subgroups in what has become known as the "Mackey machine" or "Mackey normal subgroup analysis".<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> </p> <h2 id="abstract-algebraic-setting">Abstract algebraic setting</h2> <p class="note">Main article: <a href="/facts/Frobenius_reciprocity/qoLTxRCX">Frobenius reciprocity</a></p> <p>From the point of view of <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, restriction is an instance of a <a href="/facts/Forgetful_functor/onEwu6HA">forgetful functor</a>. This functor is <a href="/facts/Exact_functor/4tsQzwBp">exact</a>, and its <a href="/facts/Adjoint_functor/bJ1P7AN2">left adjoint functor</a> is called <i>induction</i>. 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 <a href="/facts/Completely_reducible/Wz7E1GbT">complete reducibility</a>, for example, in <a href="/facts/Representation_theory_of_finite_groups/WNnRPRAK">representation theory of finite groups</a> over a <a href="/facts/Field_(algebra)/xAjAS4ko">field</a> of <a href="/facts/Characteristic_zero/D2VzcQaG">characteristic zero</a>. </p> <h2 id="generalizations">Generalizations</h2> <p>This rather evident construction may be extended in numerous and significant ways. For instance we may take any group homomorphism φ from <i>H</i> to <i>G</i>, instead of the <a href="/facts/Inclusion_map/VUFNdaDH">inclusion map</a>, and define the restricted representation of <i>H</i> by the composition </p> ρ ∘ φ {\displaystyle \rho \circ \varphi \,} <p>We may also apply the idea to other categories in <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>: <a href="/facts/Associative_algebra/DRfoF6KV">associative algebras</a>, rings, <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a>, <a href="/facts/Lie_superalgebra/c1oxUoEl">Lie superalgebras</a>, Hopf algebras to name some. Representations or <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> <i>restrict</i> to subobjects, or via homomorphisms. </p> <h2 id="notes">Notes</h2> <ul><li>Goodman, Roe; Wallach, Nolan (1998), <i>Representations and Invariants of the Classical Groups</i>, Encyclopedia Math. Appl., vol. 68, Cambridge University Press</li> <li>Helgason, Sigurdur (1978), <i>Differential geometry, Lie groups and symmetric spaces</i>, Academic Press</li> <li>Helgason, Sigurdur (1984), <a href="https://archive.org/details/groupsgeometrica0000helg"><i>Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions</i></a>, Pure and Applied Mathematics, vol. 113, Academic Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-338301-3</li> <li><a href="/facts/Roger_Evans_Howe/DQzbE2YJ">Howe, Roger</a> (1995), <i>Perspectives on invariant theory, The Schur Lectures, 1992</i>, Israel Math. Conf. Proc., vol. 8, American Mathematical Society, pp. 1–182</li> <li><a href="/facts/Roger_Evans_Howe/DQzbE2YJ">Howe, Roger</a>; Tan, Eng-Chye; <a href="/facts/Jeb_Willenbring/uEtvlLu1">Willenbring, Jeb F.</a> (2005), "Stable branching rules for classical symmetric pairs", <i>Trans. Amer. Math. Soc.</i>, 357 (4): 1601–1626, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9947-04-03722-5">10.1090/S0002-9947-04-03722-5</a></li> <li><a href="/facts/Hua_Loo-keng/g6TDG442">Hua, L.K.</a> (1963), <i>Harmonic analysis of functions of several complex variables in the classical domains</i>, American Mathematical Society</li> <li><a href="/facts/Anthony_W._Knapp/Jal9nngl">Knapp, Anthony W.</a> (2003), "Geometric interpretations of two branching theorems of D. E. 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(1973), Compact Lie groups and their representations, Translations of Mathematical Monographs, vol. 40, American Mathematical Society <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Helgason 1978 - Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Hua 1963 - Hua, L.K. (1963), Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Helgason 1984, pp. 534–543 - Helgason, Sigurdur (1984), Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, Pure and Applied Mathematics, vol. 113, Academic Press, ISBN 978-0-12-338301-3 <a href="https://archive.org/details/groupsgeometrica0000helg" target="_blank">https://archive.org/details/groupsgeometrica0000helg</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Goodman & Wallach 1998 - Goodman, Roe; Wallach, Nolan (1998), Representations and Invariants of the Classical Groups, Encyclopedia Math. Appl., vol. 68, Cambridge University Press <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Macdonald 1979 - Macdonald, Ian G. (1979), Symmetric Functions and Hall Polynomials, Oxford University Press <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Weyl 1946, p. 218 - Weyl, Hermann (1946), The classical groups, Princeton University Press <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Goodman & Wallach 1998, pp. 351–352, 365–370 - Goodman, Roe; Wallach, Nolan (1998), Representations and Invariants of the Classical Groups, Encyclopedia Math. Appl., vol. 68, Cambridge University Press <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Littlewood 1950 - Littlewood, Dudley E. (1950), The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Weyl 1946, pp. 216–222 - Weyl, Hermann (1946), The classical groups, Princeton University Press <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Koike & Terada 1987 - Koike, Kazuhiko; Terada, Itaru (1987), "Young-diagrammatic methods for the representation theory of the classical groups of type Bn, Cn, Dn", Journal of Algebra, 107 (2): 466–511, doi:10.1016/0021-8693(87)90099-8 <a href="https://doi.org/10.1016%2F0021-8693%2887%2990099-8" target="_blank">https://doi.org/10.1016%2F0021-8693%2887%2990099-8</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Macdonald 1979, p. 46 - Macdonald, Ian G. (1979), Symmetric Functions and Hall Polynomials, Oxford University Press <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Macdonald 1979 - Macdonald, Ian G. (1979), Symmetric Functions and Hall Polynomials, Oxford University Press <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Weyl 1946 - Weyl, Hermann (1946), The classical groups, Princeton University Press <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Goodman & Wallach 1998 - Goodman, Roe; Wallach, Nolan (1998), Representations and Invariants of the Classical Groups, Encyclopedia Math. Appl., vol. 68, Cambridge University Press <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Littlewood 1950, pp. 223–263 - Littlewood, Dudley E. (1950), The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Murnaghan 1938 - Murnaghan, Francis D. (1938), The Theory of Group Representations, Johns Hopkins Press <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Goodman & Wallach 1998, p. 351 - Goodman, Roe; Wallach, Nolan (1998), Representations and Invariants of the Classical Groups, Encyclopedia Math. Appl., vol. 68, Cambridge University Press <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>G. I. Olshanski had shown that the twisted Yangian Y − ( g l 2 ) {\displaystyle Y^{-}({\mathfrak {gl}}_{2})} , a sub-Hopf algebra of Y ( g l 2 ) {\displaystyle Y({\mathfrak {gl}}_{2})} , 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 g l {\displaystyle {\mathfrak {gl}}} 2. These extend to the Yangian Y ( g l ) {\displaystyle Y({\mathfrak {gl}})} and give a representation theoretic explanation of the product form of the branching coefficients. <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Weyl 1946, pp. 159–160, 311 - Weyl, Hermann (1946), The classical groups, Princeton University Press <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Mackey, George W. (1976), The theory of unitary group representations, Chicago Lectures in Mathematics, ISBN 978-0-226-50052-2 <a href="978-0-226-50052-2" target="_blank">978-0-226-50052-2</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> </ol>