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Representations of classical Lie groups

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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 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.

General linear group, special linear group and unitary group

Weyl's construction of tensor representations

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

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, 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.

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

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 (,)} . 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}}} . See for an interpretation as a product of n-dependent factors divided by products of hook lengths.

Case of the special linear group

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} . 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

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} )} .

Tensor products

Tensor products of finite-dimensional representations of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} are given by the following formula:

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 the case of tensor representations, 3-j symbols and 6-j symbols are known.

Orthogonal group and special orthogonal group

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

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.

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} . 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,

V ( k ) = U ( k ) V ( k 2 ) {\displaystyle V_{(k)}=U_{(k)}\oplus V_{(k-2)}}

Case of the special orthogonal group

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}}} .

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.

Dimensions

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} :

( 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} :

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}}}

Tensor products

In the stable range | μ | + | ν | [ n 2 ] {\displaystyle |\mu |+|\nu |\leq \left} , 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} . Beyond the stable range, the tensor product multiplicities become n {\displaystyle n} -dependent modifications of the Newell-Littlewood numbers. 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}{}\otimes &=++\\{}\otimes &=++\\{}\otimes &=++\\{}\otimes &=++++\\{}\otimes &=++\\{}\otimes &=+++++\\{}\otimes &=+++\\{}\otimes &=+++++\\{}\otimes &=++++++2+++\end{aligned}}}

Branching rules from the general linear group

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

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.

Symplectic group

Representations

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

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} :

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)}}}

Tensor products

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.

External links

References

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  2. Hawkes, Graham (2013-10-19). "An Elementary Proof of the Hook Content Formula". arXiv:1310.5919v2 .
  3. Binder, D. - Rychkov, S. (2020). "Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of O(N) Symmetry with Non-integer N". Journal of High Energy Physics. 2020 (4): 117. arXiv:1911.07895. Bibcode:2020JHEP...04..117B. doi:10.1007/JHEP04(2020)117.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  4. ^ N El Samra; R C King (December 1979). "Dimensions of irreducible representations of the classical Lie groups". Journal of Physics A. 12 (12): 2317–2328. doi:10.1088/0305-4470/12/12/010. ISSN 1751-8113. Zbl 0445.22020. Wikidata Q104601301.
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  7. Artamonov, Dmitry (2024-05-09). "Calculation of 6 j {\displaystyle 6j} -symbols for the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} ". arXiv:2405.05628 .
  8. ^ Hamermesh, Morton (1989). Group theory and its application to physical problems. New York: Dover Publications. ISBN 0-486-66181-4. OCLC 20218471.
  9. ^ Gao, Shiliang; Orelowitz, Gidon; Yong, Alexander (2021). "Newell-Littlewood numbers". Transactions of the American Mathematical Society. 374 (9): 6331–6366. arXiv:2005.09012v1. doi:10.1090/tran/8375. S2CID 218684561.
  10. Steven Sam (2010-01-18). "Littlewood-Richardson coefficients for classical groups". Concrete Nonsense. Archived from the original on 2019-06-18. Retrieved 2021-01-05.
  11. Kazuhiko Koike; Itaru Terada (May 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. ISSN 0021-8693. Zbl 0622.20033. Wikidata Q56443390.
  12. ^ Howe, Roger; Tan, Eng-Chye; Willenbring, Jeb F. (2005). "Stable branching rules for classical symmetric pairs". Transactions of the American Mathematical Society. 357 (4): 1601–1626. arXiv:math/0311159. doi:10.1090/S0002-9947-04-03722-5.
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