In mathematics, the finite-dimensional representations of the complex classical Lie groups G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} , S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} , O ( n , C ) {\displaystyle O(n,\mathbb {C} )} , S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} , S p ( 2 n , C ) {\displaystyle Sp(2n,\mathbb {C} )} , can be constructed using the general representation theory of semisimple Lie algebras . The groups S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} , S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} , S p ( 2 n , C ) {\displaystyle Sp(2n,\mathbb {C} )} are indeed simple Lie groups , and their finite-dimensional representations coincide[ 1] with those of their maximal compact subgroups , respectively S U ( n ) {\displaystyle SU(n)} , S O ( n ) {\displaystyle SO(n)} , S p ( n ) {\displaystyle Sp(n)} . In the classification of simple Lie algebras , the corresponding algebras are
S L ( n , C ) → A n − 1 S O ( n odd , C ) → B n − 1 2 S O ( n even , C ) → D n 2 S p ( 2 n , C ) → C n {\displaystyle {\begin{aligned}SL(n,\mathbb {C} )&\to A_{n-1}\\SO(n_{\text{odd}},\mathbb {C} )&\to B_{\frac {n-1}{2}}\\SO(n_{\text{even}},\mathbb {C} )&\to D_{\frac {n}{2}}\\Sp(2n,\mathbb {C} )&\to C_{n}\end{aligned}}} However, since the complex classical Lie groups are linear groups , their representations are tensor representations . Each irreducible representation is labelled by a Young diagram , which encodes its structure and properties.
Weyl's construction of tensor representations[ edit ] Let V = C n {\displaystyle V=\mathbb {C} ^{n}} be the defining representation of the general linear group G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} . Tensor representations are the subrepresentations of V ⊗ k {\displaystyle V^{\otimes k}} (these are sometimes called polynomial representations). The irreducible subrepresentations of V ⊗ k {\displaystyle V^{\otimes k}} are the images of V {\displaystyle V} by Schur functors S λ {\displaystyle \mathbb {S} ^{\lambda }} associated to integer partitions λ {\displaystyle \lambda } of k {\displaystyle k} into at most n {\displaystyle n} integers, i.e. to Young diagrams of size λ 1 + ⋯ + λ n = k {\displaystyle \lambda _{1}+\cdots +\lambda _{n}=k} with λ n + 1 = 0 {\displaystyle \lambda _{n+1}=0} . (If λ n + 1 > 0 {\displaystyle \lambda _{n+1}>0} then S λ ( V ) = 0 {\displaystyle \mathbb {S} ^{\lambda }(V)=0} .) Schur functors are defined using Young symmetrizers of the symmetric group S k {\displaystyle S_{k}} , which acts naturally on V ⊗ k {\displaystyle V^{\otimes k}} . We write V λ = S λ ( V ) {\displaystyle V_{\lambda }=\mathbb {S} ^{\lambda }(V)} .
The dimensions of these irreducible representations are[ 1]
dim V λ = ∏ 1 ≤ i < j ≤ n λ i − λ j + j − i j − i = ∏ ( i , j ) ∈ λ n − i + j h λ ( i , j ) {\displaystyle \dim V_{\lambda }=\prod _{1\leq i<j\leq n}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}=\prod _{(i,j)\in \lambda }{\frac {n-i+j}{h_{\lambda }(i,j)}}} where h λ ( i , j ) {\displaystyle h_{\lambda }(i,j)} is the hook length of the cell ( i , j ) {\displaystyle (i,j)} in the Young diagram λ {\displaystyle \lambda } .
The first formula for the dimension is a special case of a formula that gives the characters of representations in terms of Schur polynomials ,[ 1] χ λ ( g ) = s λ ( x 1 , … , x n ) {\displaystyle \chi _{\lambda }(g)=s_{\lambda }(x_{1},\dots ,x_{n})} where x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} are the eigenvalues of g ∈ G L ( n , C ) {\displaystyle g\in GL(n,\mathbb {C} )} . The second formula for the dimension is sometimes called Stanley's hook content formula .[ 2] Examples of tensor representations:
Tensor representation of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} Dimension Young diagram Trivial representation 1 {\displaystyle 1} ( ) {\displaystyle ()} Determinant representation 1 {\displaystyle 1} ( 1 n ) {\displaystyle (1^{n})} Defining representation V {\displaystyle V} n {\displaystyle n} ( 1 ) {\displaystyle (1)} Symmetric representation Sym k V {\displaystyle {\text{Sym}}^{k}V} ( n + k − 1 k ) {\displaystyle {\binom {n+k-1}{k}}} ( k ) {\displaystyle (k)} Antisymmetric representation Λ k V {\displaystyle \Lambda ^{k}V} ( n k ) {\displaystyle {\binom {n}{k}}} ( 1 k ) {\displaystyle (1^{k})}
General irreducible representations [ edit ] Not all irreducible representations of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} are tensor representations. In general, irreducible representations of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} are mixed tensor representations, i.e. subrepresentations of V ⊗ r ⊗ ( V ∗ ) ⊗ s {\displaystyle V^{\otimes r}\otimes (V^{*})^{\otimes s}} , where V ∗ {\displaystyle V^{*}} is the dual representation of V {\displaystyle V} (these are sometimes called rational representations). In the end, the set of irreducible representations of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} is labeled by non increasing sequences of n {\displaystyle n} integers λ 1 ≥ ⋯ ≥ λ n {\displaystyle \lambda _{1}\geq \dots \geq \lambda _{n}} . If λ k ≥ 0 , λ k + 1 ≤ 0 {\displaystyle \lambda _{k}\geq 0,\lambda _{k+1}\leq 0} , we can associate to ( λ 1 , … , λ n ) {\displaystyle (\lambda _{1},\dots ,\lambda _{n})} the pair of Young tableaux ( [ λ 1 … λ k ] , [ − λ n , … , − λ k + 1 ] ) {\displaystyle ([\lambda _{1}\dots \lambda _{k}],[-\lambda _{n},\dots ,-\lambda _{k+1}])} . This shows that irreducible representations of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} can be labeled by pairs of Young tableaux . Let us denote V λ μ = V λ 1 , … , λ n {\displaystyle V_{\lambda \mu }=V_{\lambda _{1},\dots ,\lambda _{n}}} the irreducible representation of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} corresponding to the pair ( λ , μ ) {\displaystyle (\lambda ,\mu )} or equivalently to the sequence ( λ 1 , … , λ n ) {\displaystyle (\lambda _{1},\dots ,\lambda _{n})} . With these notations,
V λ = V λ ( ) , V = V ( 1 ) ( ) {\displaystyle V_{\lambda }=V_{\lambda ()},V=V_{(1)()}} ( V λ μ ) ∗ = V μ λ {\displaystyle (V_{\lambda \mu })^{*}=V_{\mu \lambda }} For k ∈ Z {\displaystyle k\in \mathbb {Z} } , denoting D k {\displaystyle D_{k}} the one-dimensional representation in which G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} acts by ( det ) k {\displaystyle (\det )^{k}} , V λ 1 , … , λ n = V λ 1 + k , … , λ n + k ⊗ D − k {\displaystyle V_{\lambda _{1},\dots ,\lambda _{n}}=V_{\lambda _{1}+k,\dots ,\lambda _{n}+k}\otimes D_{-k}} . If k {\displaystyle k} is large enough that λ n + k ≥ 0 {\displaystyle \lambda _{n}+k\geq 0} , this gives an explicit description of V λ 1 , … , λ n {\displaystyle V_{\lambda _{1},\dots ,\lambda _{n}}} in terms of a Schur functor. The dimension of V λ μ {\displaystyle V_{\lambda \mu }} where λ = ( λ 1 , … , λ r ) , μ = ( μ 1 , … , μ s ) {\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{r}),\mu =(\mu _{1},\dots ,\mu _{s})} is dim ( V λ μ ) = d λ d μ ∏ i = 1 r ( 1 − i − s + n ) λ i ( 1 − i + r ) λ i ∏ j = 1 s ( 1 − j − r + n ) μ i ( 1 − j + s ) μ i ∏ i = 1 r ∏ j = 1 s n + 1 + λ i + μ j − i − j n + 1 − i − j {\displaystyle \dim(V_{\lambda \mu })=d_{\lambda }d_{\mu }\prod _{i=1}^{r}{\frac {(1-i-s+n)_{\lambda _{i}}}{(1-i+r)_{\lambda _{i}}}}\prod _{j=1}^{s}{\frac {(1-j-r+n)_{\mu _{i}}}{(1-j+s)_{\mu _{i}}}}\prod _{i=1}^{r}\prod _{j=1}^{s}{\frac {n+1+\lambda _{i}+\mu _{j}-i-j}{n+1-i-j}}} where d λ = ∏ 1 ≤ i < j ≤ r λ i − λ j + j − i j − i {\displaystyle d_{\lambda }=\prod _{1\leq i<j\leq r}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}} .[ 3] See [ 4] for an interpretation as a product of n-dependent factors divided by products of hook lengths. Case of the special linear group [ edit ] Two representations V λ , V λ ′ {\displaystyle V_{\lambda },V_{\lambda '}} of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} are equivalent as representations of the special linear group S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} if and only if there is k ∈ Z {\displaystyle k\in \mathbb {Z} } such that ∀ i , λ i − λ i ′ = k {\displaystyle \forall i,\ \lambda _{i}-\lambda '_{i}=k} .[ 1] For instance, the determinant representation V ( 1 n ) {\displaystyle V_{(1^{n})}} is trivial in S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} , i.e. it is equivalent to V ( ) {\displaystyle V_{()}} . In particular, irreducible representations of S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} can be indexed by Young tableaux, and are all tensor representations (not mixed).
Case of the unitary group [ edit ] The unitary group is the maximal compact subgroup of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} . The complexification of its Lie algebra u ( n ) = { a ∈ M ( n , C ) , a † + a = 0 } {\displaystyle {\mathfrak {u}}(n)=\{a\in {\mathcal {M}}(n,\mathbb {C} ),a^{\dagger }+a=0\}} is the algebra g l ( n , C ) {\displaystyle {\mathfrak {gl}}(n,\mathbb {C} )} . In Lie theoretic terms, U ( n ) {\displaystyle U(n)} is the compact real form of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} , which means that complex linear, continuous irreducible representations of the latter are in one-to-one correspondence with complex linear, algebraic irreps of the former, via the inclusion U ( n ) → G L ( n , C ) {\displaystyle U(n)\rightarrow GL(n,\mathbb {C} )} . [ 5]
Tensor products of finite-dimensional representations of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} are given by the following formula:[ 6]
V λ 1 μ 1 ⊗ V λ 2 μ 2 = ⨁ ν , ρ V ν ρ ⊕ Γ λ 1 μ 1 , λ 2 μ 2 ν ρ , {\displaystyle V_{\lambda _{1}\mu _{1}}\otimes V_{\lambda _{2}\mu _{2}}=\bigoplus _{\nu ,\rho }V_{\nu \rho }^{\oplus \Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }},} where Γ λ 1 μ 1 , λ 2 μ 2 ν ρ = 0 {\displaystyle \Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }=0} unless | ν | ≤ | λ 1 | + | λ 2 | {\displaystyle |\nu |\leq |\lambda _{1}|+|\lambda _{2}|} and | ρ | ≤ | μ 1 | + | μ 2 | {\displaystyle |\rho |\leq |\mu _{1}|+|\mu _{2}|} . Calling l ( λ ) {\displaystyle l(\lambda )} the number of lines in a tableau, if l ( λ 1 ) + l ( λ 2 ) + l ( μ 1 ) + l ( μ 2 ) ≤ n {\displaystyle l(\lambda _{1})+l(\lambda _{2})+l(\mu _{1})+l(\mu _{2})\leq n} , then
Γ λ 1 μ 1 , λ 2 μ 2 ν ρ = ∑ α , β , η , θ ( ∑ κ c κ , α λ 1 c κ , β μ 2 ) ( ∑ γ c γ , η λ 2 c γ , θ μ 1 ) c α , θ ν c β , η ρ , {\displaystyle \Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }=\sum _{\alpha ,\beta ,\eta ,\theta }\left(\sum _{\kappa }c_{\kappa ,\alpha }^{\lambda _{1}}c_{\kappa ,\beta }^{\mu _{2}}\right)\left(\sum _{\gamma }c_{\gamma ,\eta }^{\lambda _{2}}c_{\gamma ,\theta }^{\mu _{1}}\right)c_{\alpha ,\theta }^{\nu }c_{\beta ,\eta }^{\rho },} where the natural integers c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }} are Littlewood-Richardson coefficients .
Below are a few examples of such tensor products:
R 1 {\displaystyle R_{1}} R 2 {\displaystyle R_{2}} Tensor product R 1 ⊗ R 2 {\displaystyle R_{1}\otimes R_{2}} V λ ( ) {\displaystyle V_{\lambda ()}} V μ ( ) {\displaystyle V_{\mu ()}} ∑ ν c λ μ ν V ν ( ) {\displaystyle \sum _{\nu }c_{\lambda \mu }^{\nu }V_{\nu ()}} V λ ( ) {\displaystyle V_{\lambda ()}} V ( ) μ {\displaystyle V_{()\mu }} ∑ κ , ν , ρ c κ ν λ c κ ρ μ V ν ρ {\displaystyle \sum _{\kappa ,\nu ,\rho }c_{\kappa \nu }^{\lambda }c_{\kappa \rho }^{\mu }V_{\nu \rho }} V ( ) ( 1 ) {\displaystyle V_{()(1)}} V ( 1 ) ( ) {\displaystyle V_{(1)()}} V ( 1 ) ( 1 ) + V ( ) ( ) {\displaystyle V_{(1)(1)}+V_{()()}} V ( ) ( 1 ) {\displaystyle V_{()(1)}} V ( k ) ( ) {\displaystyle V_{(k)()}} V ( k ) ( 1 ) + V ( k − 1 ) ( ) {\displaystyle V_{(k)(1)}+V_{(k-1)()}} V ( 1 ) ( ) {\displaystyle V_{(1)()}} V ( k ) ( ) {\displaystyle V_{(k)()}} V ( k + 1 ) ( ) + V ( k , 1 ) ( ) {\displaystyle V_{(k+1)()}+V_{(k,1)()}} V ( 1 ) ( 1 ) {\displaystyle V_{(1)(1)}} V ( 1 ) ( 1 ) {\displaystyle V_{(1)(1)}} V ( 2 ) ( 2 ) + V ( 2 ) ( 11 ) + V ( 11 ) ( 2 ) + V ( 11 ) ( 11 ) + 2 V ( 1 ) ( 1 ) + V ( ) ( ) {\displaystyle V_{(2)(2)}+V_{(2)(11)}+V_{(11)(2)}+V_{(11)(11)}+2V_{(1)(1)}+V_{()()}}
In addition to the Lie group representations described here, the orthogonal group O ( n , C ) {\displaystyle O(n,\mathbb {C} )} and special orthogonal group S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} have spin representations , which are projective representations of these groups, i.e. representations of their universal covering groups .
Construction of representations [ edit ] Since O ( n , C ) {\displaystyle O(n,\mathbb {C} )} is a subgroup of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} , any irreducible representation of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} is also a representation of O ( n , C ) {\displaystyle O(n,\mathbb {C} )} , which may however not be irreducible. In order for a tensor representation of O ( n , C ) {\displaystyle O(n,\mathbb {C} )} to be irreducible, the tensors must be traceless.[ 7]
Irreducible representations of O ( n , C ) {\displaystyle O(n,\mathbb {C} )} are parametrized by a subset of the Young diagrams associated to irreducible representations of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} : the diagrams such that the sum of the lengths of the first two columns is at most n {\displaystyle n} .[ 7] The irreducible representation U λ {\displaystyle U_{\lambda }} that corresponds to such a diagram is a subrepresentation of the corresponding G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} representation V λ {\displaystyle V_{\lambda }} . For example, in the case of symmetric tensors,[ 1]
V ( k ) = U ( k ) ⊕ V ( k − 2 ) {\displaystyle V_{(k)}=U_{(k)}\oplus V_{(k-2)}} Case of the special orthogonal group [ edit ] The antisymmetric tensor U ( 1 n ) {\displaystyle U_{(1^{n})}} is a one-dimensional representation of O ( n , C ) {\displaystyle O(n,\mathbb {C} )} , which is trivial for S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} . Then U ( 1 n ) ⊗ U λ = U λ ′ {\displaystyle U_{(1^{n})}\otimes U_{\lambda }=U_{\lambda '}} where λ ′ {\displaystyle \lambda '} is obtained from λ {\displaystyle \lambda } by acting on the length of the first column as λ ~ 1 → n − λ ~ 1 {\displaystyle {\tilde {\lambda }}_{1}\to n-{\tilde {\lambda }}_{1}} .
For n {\displaystyle n} odd, the irreducible representations of S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} are parametrized by Young diagrams with λ ~ 1 ≤ n − 1 2 {\displaystyle {\tilde {\lambda }}_{1}\leq {\frac {n-1}{2}}} rows. For n {\displaystyle n} even, U λ {\displaystyle U_{\lambda }} is still irreducible as an S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} representation if λ ~ 1 ≤ n 2 − 1 {\displaystyle {\tilde {\lambda }}_{1}\leq {\frac {n}{2}}-1} , but it reduces to a sum of two inequivalent S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} representations if λ ~ 1 = n 2 {\displaystyle {\tilde {\lambda }}_{1}={\frac {n}{2}}} .[ 7] For example, the irreducible representations of O ( 3 , C ) {\displaystyle O(3,\mathbb {C} )} correspond to Young diagrams of the types ( k ≥ 0 ) , ( k ≥ 1 , 1 ) , ( 1 , 1 , 1 ) {\displaystyle (k\geq 0),(k\geq 1,1),(1,1,1)} . The irreducible representations of S O ( 3 , C ) {\displaystyle SO(3,\mathbb {C} )} correspond to ( k ≥ 0 ) {\displaystyle (k\geq 0)} , and dim U ( k ) = 2 k + 1 {\displaystyle \dim U_{(k)}=2k+1} . On the other hand, the dimensions of the spin representations of S O ( 3 , C ) {\displaystyle SO(3,\mathbb {C} )} are even integers.[ 1]
The dimensions of irreducible representations of S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} are given by a formula that depends on the parity of n {\displaystyle n} :[ 4]
( n even ) dim U λ = ∏ 1 ≤ i < j ≤ n 2 λ i − λ j − i + j − i + j ⋅ λ i + λ j + n − i − j n − i − j {\displaystyle (n{\text{ even}})\qquad \dim U_{\lambda }=\prod _{1\leq i<j\leq {\frac {n}{2}}}{\frac {\lambda _{i}-\lambda _{j}-i+j}{-i+j}}\cdot {\frac {\lambda _{i}+\lambda _{j}+n-i-j}{n-i-j}}} ( n odd ) dim U λ = ∏ 1 ≤ i < j ≤ n − 1 2 λ i − λ j − i + j − i + j ∏ 1 ≤ i ≤ j ≤ n − 1 2 λ i + λ j + n − i − j n − i − j {\displaystyle (n{\text{ odd}})\qquad \dim U_{\lambda }=\prod _{1\leq i<j\leq {\frac {n-1}{2}}}{\frac {\lambda _{i}-\lambda _{j}-i+j}{-i+j}}\prod _{1\leq i\leq j\leq {\frac {n-1}{2}}}{\frac {\lambda _{i}+\lambda _{j}+n-i-j}{n-i-j}}} There is also an expression as a factorized polynomial in n {\displaystyle n} :[ 4]
dim U λ = ∏ ( i , j ) ∈ λ , i ≥ j n + λ i + λ j − i − j h λ ( i , j ) ∏ ( i , j ) ∈ λ , i < j n − λ ~ i − λ ~ j + i + j − 2 h λ ( i , j ) {\displaystyle \dim U_{\lambda }=\prod _{(i,j)\in \lambda ,\ i\geq j}{\frac {n+\lambda _{i}+\lambda _{j}-i-j}{h_{\lambda }(i,j)}}\prod _{(i,j)\in \lambda ,\ i<j}{\frac {n-{\tilde {\lambda }}_{i}-{\tilde {\lambda }}_{j}+i+j-2}{h_{\lambda }(i,j)}}} where λ i , λ ~ i , h λ ( i , j ) {\displaystyle \lambda _{i},{\tilde {\lambda }}_{i},h_{\lambda }(i,j)} are respectively row lengths, column lengths and hook lengths . In particular, antisymmetric representations have the same dimensions as their G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} counterparts, dim U ( 1 k ) = dim V ( 1 k ) {\displaystyle \dim U_{(1^{k})}=\dim V_{(1^{k})}} , but symmetric representations do not,
dim U ( k ) = dim V ( k ) − dim V ( k − 2 ) = ( n + k − 1 k ) − ( n + k − 3 k ) {\displaystyle \dim U_{(k)}=\dim V_{(k)}-\dim V_{(k-2)}={\binom {n+k-1}{k}}-{\binom {n+k-3}{k}}} In the stable range | μ | + | ν | ≤ [ n 2 ] {\displaystyle |\mu |+|\nu |\leq \left[{\frac {n}{2}}\right]} , the tensor product multiplicities that appear in the tensor product decomposition U λ ⊗ U μ = ⊕ ν N λ , μ , ν U ν {\displaystyle U_{\lambda }\otimes U_{\mu }=\oplus _{\nu }N_{\lambda ,\mu ,\nu }U_{\nu }} are Newell-Littlewood numbers , which do not depend on n {\displaystyle n} .[ 8] Beyond the stable range, the tensor product multiplicities become n {\displaystyle n} -dependent modifications of the Newell-Littlewood numbers.[ 9] [ 8] [ 10] For example, for n ≥ 12 {\displaystyle n\geq 12} , we have
[ 1 ] ⊗ [ 1 ] = [ 2 ] + [ 11 ] + [ ] [ 1 ] ⊗ [ 2 ] = [ 21 ] + [ 3 ] + [ 1 ] [ 1 ] ⊗ [ 11 ] = [ 111 ] + [ 21 ] + [ 1 ] [ 1 ] ⊗ [ 21 ] = [ 31 ] + [ 22 ] + [ 211 ] + [ 2 ] + [ 11 ] [ 1 ] ⊗ [ 3 ] = [ 4 ] + [ 31 ] + [ 2 ] [ 2 ] ⊗ [ 2 ] = [ 4 ] + [ 31 ] + [ 22 ] + [ 2 ] + [ 11 ] + [ ] [ 2 ] ⊗ [ 11 ] = [ 31 ] + [ 211 ] + [ 2 ] + [ 11 ] [ 11 ] ⊗ [ 11 ] = [ 1111 ] + [ 211 ] + [ 22 ] + [ 2 ] + [ 11 ] + [ ] [ 21 ] ⊗ [ 3 ] = [ 321 ] + [ 411 ] + [ 42 ] + [ 51 ] + [ 211 ] + [ 22 ] + 2 [ 31 ] + [ 4 ] + [ 11 ] + [ 2 ] {\displaystyle {\begin{aligned}{}[1]\otimes [1]&=[2]+[11]+[]\\{}[1]\otimes [2]&=[21]+[3]+[1]\\{}[1]\otimes [11]&=[111]+[21]+[1]\\{}[1]\otimes [21]&=[31]+[22]+[211]+[2]+[11]\\{}[1]\otimes [3]&=[4]+[31]+[2]\\{}[2]\otimes [2]&=[4]+[31]+[22]+[2]+[11]+[]\\{}[2]\otimes [11]&=[31]+[211]+[2]+[11]\\{}[11]\otimes [11]&=[1111]+[211]+[22]+[2]+[11]+[]\\{}[21]\otimes [3]&=[321]+[411]+[42]+[51]+[211]+[22]+2[31]+[4]+[11]+[2]\end{aligned}}} Branching rules from the general linear group [ edit ] Since the orthogonal group is a subgroup of the general linear group, representations of G L ( n ) {\displaystyle GL(n)} can be decomposed into representations of O ( n ) {\displaystyle O(n)} . The decomposition of a tensor representation is given in terms of Littlewood-Richardson coefficients c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }} by the Littlewood restriction rule[ 11]
V ν G L ( n ) = ∑ λ , μ c λ , 2 μ ν U λ O ( n ) {\displaystyle V_{\nu }^{GL(n)}=\sum _{\lambda ,\mu }c_{\lambda ,2\mu }^{\nu }U_{\lambda }^{O(n)}} where 2 μ {\displaystyle 2\mu } is a partition into even integers. The rule is valid in the stable range 2 | ν | , λ ~ 1 + λ ~ 2 ≤ n {\displaystyle 2|\nu |,{\tilde {\lambda }}_{1}+{\tilde {\lambda }}_{2}\leq n} . The generalization to mixed tensor representations is
V λ μ G L ( n ) = ∑ α , β , γ , δ c α , 2 γ λ c β , 2 δ μ c α , β ν U ν O ( n ) {\displaystyle V_{\lambda \mu }^{GL(n)}=\sum _{\alpha ,\beta ,\gamma ,\delta }c_{\alpha ,2\gamma }^{\lambda }c_{\beta ,2\delta }^{\mu }c_{\alpha ,\beta }^{\nu }U_{\nu }^{O(n)}} Similar branching rules can be written for the symplectic group.[ 11]
The finite-dimensional irreducible representations of the symplectic group S p ( 2 n , C ) {\displaystyle Sp(2n,\mathbb {C} )} are parametrized by Young diagrams with at most n {\displaystyle n} rows. The dimension of the corresponding representation is[ 7]
dim W λ = ∏ i = 1 n λ i + n − i + 1 n − i + 1 ∏ 1 ≤ i < j ≤ n λ i − λ j + j − i j − i ⋅ λ i + λ j + 2 n − i − j + 2 2 n − i − j + 2 {\displaystyle \dim W_{\lambda }=\prod _{i=1}^{n}{\frac {\lambda _{i}+n-i+1}{n-i+1}}\prod _{1\leq i<j\leq n}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}\cdot {\frac {\lambda _{i}+\lambda _{j}+2n-i-j+2}{2n-i-j+2}}} There is also an expression as a factorized polynomial in n {\displaystyle n} :[ 4]
dim W λ = ∏ ( i , j ) ∈ λ , i > j n + λ i + λ j − i − j + 2 h λ ( i , j ) ∏ ( i , j ) ∈ λ , i ≤ j n − λ ~ i − λ ~ j + i + j h λ ( i , j ) {\displaystyle \dim W_{\lambda }=\prod _{(i,j)\in \lambda ,\ i>j}{\frac {n+\lambda _{i}+\lambda _{j}-i-j+2}{h_{\lambda }(i,j)}}\prod _{(i,j)\in \lambda ,\ i\leq j}{\frac {n-{\tilde {\lambda }}_{i}-{\tilde {\lambda }}_{j}+i+j}{h_{\lambda }(i,j)}}} Just like in the case of the orthogonal group, tensor product multiplicities are given by Newell-Littlewood numbers in the stable range, and modifications thereof beyond the stable range.
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