View source for Representations of classical Lie groups

{{Lie groups|Representation}}

In mathematics, the finite-dimensional [[representation theory|representations]] of the complex [[classical Lie groups]] 
<math>GL(n,\mathbb{C})</math>, <math>SL(n,\mathbb{C})</math>, <math>O(n,\mathbb{C})</math>, <math>SO(n,\mathbb{C})</math>, <math>Sp(2n,\mathbb{C})</math>,
can be constructed using the general [[representation theory of semisimple Lie algebras]]. The groups 
<math>SL(n,\mathbb{C})</math>, <math>SO(n,\mathbb{C})</math>, <math>Sp(2n,\mathbb{C})</math> are indeed [[simple Lie groups]], and their finite-dimensional representations coincide<ref name = "Fulton">{{cite Q|Q55865630}}</ref> with those of their [[maximal compact subgroup]]s, respectively <math>SU(n)</math>, <math>SO(n)</math>, <math>Sp(n)</math>. In the [[Semisimple Lie algebra#Classification|classification of simple Lie algebras]], the corresponding algebras are 
:<math>
\begin{align}
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{align}
</math>
However, since the complex classical Lie groups are [[linear group]]s, their representations are [[tensor representation]]s. 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 <math>V=\mathbb{C}^n</math> be the defining representation of the general linear group <math>GL(n,\mathbb{C})</math>. [[Tensor representations]] are the subrepresentations of <math>V^{\otimes k}</math> (these are sometimes called polynomial representations). The irreducible subrepresentations of <math>V^{\otimes k}</math> are the images of <math>V</math> by [[Schur functor]]s <math>\mathbb{S}^\lambda</math> associated to [[integer partition]]s <math>\lambda</math> of <math>k</math> into at most <math>n</math> integers, i.e. to [[Young diagrams]] of size <math>\lambda_1+\cdots + \lambda_n = k</math> with <math>\lambda_{n+1}=0</math>. (If <math>\lambda_{n+1}>0</math> then <math>\mathbb{S}^\lambda(V)=0</math>.) Schur functors are defined using [[Young symmetrizer]]s of the [[symmetric group]] <math>S_k</math>, which acts naturally on <math>V^{\otimes k}</math>. We write <math>V_\lambda = \mathbb{S}^\lambda(V)</math>.

The dimensions of these irreducible representations are<ref name = "Fulton">{{cite Q|Q55865630}}</ref>
:<math>
\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)}
</math> 
where <math>h_\lambda(i,j)</math> is the [[Hook length formula#Definitions and statement|hook length]] of the cell <math>(i,j)</math> in the Young diagram <math>\lambda</math>. 
* The first formula for the dimension is a special case of a formula that gives the [[character theory|characters]] of representations in terms of [[Schur polynomials]],<ref name = "Fulton">{{cite Q|Q55865630}}</ref> <math>
\chi_\lambda(g) = s_\lambda(x_1,\dots, x_n)
</math> where <math>x_1,\dots ,x_n</math> are the eigenvalues of <math>g\in GL(n,\mathbb{C})</math>. 
* The second formula for the dimension is sometimes called '''Stanley's hook content formula'''.<ref name="haw13"/>

Examples of tensor representations:

{| class="wikitable" style="font-size:small; text-align:center;"
|-
! Tensor representation of <math>GL(n,\mathbb{C})</math>
! Dimension
! [[Young diagram]]
|-
| Trivial representation
| <math>1 </math>
| <math> () </math>
|-
| Determinant representation
| <math>1</math> 
| <math>(1^n)</math>
|-
| Defining representation <math>V</math>
| <math>n</math>
| <math>(1)</math>
|-
| Symmetric representation <math> \text{Sym}^kV</math>
| <math>\binom{n+k-1}{k}</math>
| <math>(k)</math>
|- 
| Antisymmetric representation <math> \Lambda^k V</math> 
| <math>\binom{n}{k}</math> 
| <math> (1^k)</math> 
|}

=== General irreducible representations ===

Not all irreducible representations of <math> GL(n,\mathbb C) </math> are tensor representations. In general, irreducible representations of <math> GL(n,\mathbb C) </math> are mixed tensor representations, i.e. subrepresentations of <math> V^{\otimes r} \otimes (V^*)^{\otimes s}</math>, where <math> V^* </math> is the [[dual representation]] of <math> V </math> (these are sometimes called rational representations). In the end, the set of irreducible representations of <math> GL(n,\mathbb C)</math>  is labeled by non increasing sequences of <math> n </math> integers <math> \lambda_1\geq \dots \geq \lambda_n </math>.
If <math> \lambda_k \geq 0, \lambda_{k+1} \leq 0 </math>, we can associate to <math> (\lambda_1, \dots ,\lambda_n) </math> the pair of Young tableaux <math> ([\lambda_1\dots\lambda_k],[-\lambda_n,\dots,-\lambda_{k+1}]) </math>. This shows that irreducible representations of <math> GL(n,\mathbb C) </math> can be labeled by pairs of Young tableaux . Let us denote <math> V_{\lambda\mu} = V_{\lambda_1,\dots,\lambda_n} </math> the irreducible representation of <math> GL(n,\mathbb C) </math> corresponding to the pair <math>(\lambda,\mu)</math> or equivalently to the sequence <math> (\lambda_1,\dots,\lambda_n) </math>. With these notations,

* <math>V_{\lambda}=V_{\lambda()}, V = V_{(1)()}</math>

* <math> (V_{\lambda\mu})^* = V_{\mu\lambda} </math>

* For <math> k \in \mathbb Z </math>, denoting <math> D_k </math> the one-dimensional representation in which <math> GL(n,\mathbb C) </math> acts by <math> (\det)^k </math>, <math> V_{\lambda_1,\dots,\lambda_n} = V_{\lambda_1+k,\dots,\lambda_n+k} \otimes D_{-k} </math>. If <math> k </math> is large enough that <math> \lambda_n + k \geq 0 </math>, this gives an explicit description of <math> V_{\lambda_1, \dots,\lambda_n} </math> in terms of a Schur functor.

* The dimension of <math> V_{\lambda\mu} </math>  where <math> \lambda = (\lambda_1,\dots,\lambda_r), \mu=(\mu_1,\dots,\mu_s) </math> is 
:<math> \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 }
</math> where <math> d_\lambda = \prod_{1 \leq i < j \leq r} \frac{\lambda_i - \lambda_j + j - i}{j-i} </math>.<ref>{{cite journal | last1=Binder, D. - Rychkov, S. |title=Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of O(N) Symmetry with Non-integer N |journal=Journal of High Energy Physics |year=2020 |volume=2020 |issue=4 |page=117 |doi=10.1007/JHEP04(2020)117|arxiv=1911.07895 |bibcode=2020JHEP...04..117B |doi-access=free }}</ref> See <ref name="El Samra" /> for an interpretation as a product of n-dependent factors divided by products of hook lengths.

=== Case of the special linear group ===

Two representations <math>V_{\lambda},V_{\lambda'}</math> of <math>GL(n,\mathbb{C})</math> are equivalent as representations of the special linear group <math>SL(n,\mathbb{C})</math> if and only if there is <math>k\in\mathbb{Z}</math> such that <math>\forall i,\ \lambda_i-\lambda'_i=k</math>.<ref name = "Fulton">{{cite Q|Q55865630}}</ref> For instance, the determinant representation <math>V_{(1^n)}</math> is trivial in <math>SL(n,\mathbb{C})</math>, i.e. it is equivalent to <math>V_{()}</math>. 
In particular, irreducible representations of <math> SL(n,\mathbb C) </math> 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 <math> GL(n,\mathbb C) </math>. The complexification of its Lie algebra <math>\mathfrak u(n) = \{a \in \mathcal M(n,\mathbb C), a^\dagger + a = 0\}</math>  is the algebra <math>\mathfrak{gl}(n,\mathbb C)</math>. In Lie theoretic terms, <math> U(n) </math> is the compact [[real form]] of <math> GL(n,\mathbb C) </math>, 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 <math> U(n) \rightarrow GL(n,\mathbb C) </math>. <ref>{{cite book |last1=Cvitanović |first1=Predrag |title=Group theory: Birdtracks, Lie's, and exceptional groups |date=2008 |url=https://birdtracks.eu/}}</ref>

=== Tensor products ===

Tensor products of finite-dimensional representations of <math>GL(n,\mathbb{C})</math> are given by the following formula:<ref>{{cite journal |last1=Koike |first1=Kazuhiko |title=On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters |journal=[[Advances in Mathematics]] |year=1989 |volume=74 |pages=57–86 |doi=10.1016/0001-8708(89)90004-2|doi-access=free }}</ref>

:<math> 
V_{\lambda_1\mu_1} \otimes V_{\lambda_2\mu_2} = \bigoplus_{\nu,\rho} V_{\nu\rho}^{\oplus \Gamma^{\nu\rho}_{\lambda_1\mu_1,\lambda_2\mu_2}},
</math>
where <math> \Gamma^{\nu\rho}_{\lambda_1\mu_1,\lambda_2\mu_2} = 0 </math> unless <math> |\nu| \leq |\lambda_1| + |\lambda_2|</math> and <math> |\rho| \leq |\mu_1| + |\mu_2|</math>. Calling <math> l(\lambda)</math> the number of lines in a tableau, if <math> l(\lambda_1) + l(\lambda_2) + l(\mu_1) + l(\mu_2) \leq n </math>, then 
:<math>
 \Gamma^{\nu\rho}_{\lambda_1\mu_1,\lambda_2\mu_2} = \sum_{\alpha,\beta,\eta,\theta} \left(\sum_\kappa c^{\lambda_1}_{\kappa,\alpha} c^{\mu_2}_{\kappa,\beta}\right)\left(\sum_\gamma c^{\lambda_2}_{\gamma,\eta}c^{\mu_1}_{\gamma,\theta}\right)c^{\nu}_{\alpha,\theta}c^{\rho}_{\beta,\eta},
</math>
where the natural integers <math>c_{\lambda,\mu}^\nu</math> are 
[[Littlewood-Richardson coefficient]]s.

Below are a few examples of such tensor products:

{| class="wikitable" style="font-size:small; text-align:left;"
|-
! <math> R_1 </math>
! <math> R_2 </math>
! Tensor product <math> R_1 \otimes R_2 </math>
|-
| <math> V_{\lambda()} </math>
| <math> V_{\mu()} </math>
| <math> \sum_\nu c^\nu_{\lambda \mu}V_{\nu()} </math>
|-
| <math> V_{\lambda()} </math>
| <math> V_{()\mu} </math>
| <math> \sum_{\kappa,\nu,\rho} c^\lambda_{\kappa\nu} c^{\mu}_{\kappa\rho} V_{\nu\rho} </math>
|-
| <math> V_{()(1)} </math>
| <math> V_{(1)()} </math>
| <math> V_{(1)(1)} + V_{()()} </math>
|-
| <math> V_{()(1)} </math>
| <math> V_{(k)()} </math>
| <math> V_{(k)(1)} + V_{(k-1)()} </math>
|-
| <math> V_{(1)()} </math>
| <math> V_{(k)()} </math>
| <math> V_{(k+1)()} + V_{(k,1)()} </math>
|-
| <math> V_{(1)(1)} </math>
| <math> V_{(1)(1)} </math>
| <math> V_{(2)(2)} + V_{(2)(11)} + V_{(11)(2)} + V_{(11)(11)} + 2V_{(1)(1)} + V_{()()} </math>
|}
In the case of tensor representations, [[3-j symbol]]s and [[6-j symbol]]s are known.<ref name="art24"/>

== [[Orthogonal group]] and [[special orthogonal group]] ==

''In addition to the [[Lie group representation]]s described here, the orthogonal group <math>O(n,\mathbb{C})</math> and special orthogonal group <math>SO(n,\mathbb{C})</math> have [[spin representations]], which are [[projective representation]]s of these groups, i.e. representations of their [[Covering group#Universal covering group|universal covering groups]].''

=== Construction of representations ===

Since <math>O(n,\mathbb{C})</math> is a subgroup of <math>GL(n,\mathbb{C})</math>, any irreducible representation of <math>GL(n,\mathbb{C})</math> is also a representation of <math>O(n,\mathbb{C})</math>, which may however not be irreducible. In order for a tensor representation of <math>O(n,\mathbb{C})</math> to be irreducible, the tensors must be traceless.<ref name="Hamermesh"/>

Irreducible representations of <math>O(n,\mathbb{C})</math> are parametrized by a subset of the Young diagrams associated to irreducible representations of <math>GL(n,\mathbb{C})</math>: the diagrams such that the sum of the lengths of the first two columns is at most <math>n</math>.<ref name="Hamermesh"/> The irreducible representation <math>U_\lambda</math> that corresponds to such a diagram is a subrepresentation of the corresponding <math>GL(n,\mathbb{C})</math> representation <math>V_\lambda</math>. For example, in the case of symmetric tensors,<ref name = "Fulton">{{cite Q|Q55865630}}</ref> 
:<math>
V_{(k)} = U_{(k)} \oplus V_{(k-2)}
</math>

=== Case of the special orthogonal group ===

The antisymmetric tensor <math>U_{(1^n)}</math> is a one-dimensional representation of <math>O(n,\mathbb{C})</math>, which is trivial for <math>SO(n,\mathbb{C})</math>. Then <math>U_{(1^n)}\otimes U_\lambda = U_{\lambda'}</math> where <math>\lambda'</math> is obtained from <math>\lambda</math> by acting on the length of the first column as <math>\tilde{\lambda}_1\to n-\tilde{\lambda}_1</math>. 
* For <math>n</math> odd, the irreducible representations of <math>SO(n,\mathbb{C})</math> are parametrized by Young diagrams with <math>\tilde{\lambda}_1\leq\frac{n-1}{2}</math> rows.
* For <math>n</math> even, <math>U_\lambda</math> is still irreducible as an <math>SO(n,\mathbb{C})</math> representation if <math>\tilde{\lambda}_1\leq\frac{n}{2}-1</math>, but it reduces to a sum of two inequivalent <math>SO(n,\mathbb{C})</math> representations if <math>\tilde{\lambda}_1=\frac{n}{2}</math>.<ref name="Hamermesh"/>

For example, the irreducible representations of <math>O(3,\mathbb{C})</math> correspond to Young diagrams of the types <math>(k\geq 0),(k\geq 1,1),(1,1,1)</math>. The irreducible representations of <math>SO(3,\mathbb{C})</math> correspond to <math>(k\geq 0)</math>, and <math>\dim U_{(k)}=2k+1</math>. 
On the other hand, the dimensions of the [[spin representations]] of <math>SO(3,\mathbb{C})</math> are even integers.<ref name = "Fulton">{{cite Q|Q55865630}}</ref>

=== Dimensions ===

The dimensions of irreducible representations of <math>SO(n,\mathbb{C})</math> are given by a formula that depends on the parity of <math>n</math>:<ref name="El Samra">{{cite Q|Q104601301}}</ref> 
:<math>
(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}
</math>
:<math>
(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}
</math>
There is also an expression as a factorized polynomial in <math>n</math>:<ref name="El Samra">{{cite Q|Q104601301}}</ref> 
:<math>
\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)}
</math>
where <math>\lambda_i,\tilde{\lambda}_i,h_\lambda(i,j)</math> are respectively row lengths, column lengths and [[hook length formula|hook lengths]]. In particular, antisymmetric representations have the same dimensions as their <math>GL(n,\mathbb{C})</math> counterparts, <math>\dim U_{(1^k)}=\dim V_{(1^k)}</math>, but symmetric representations do not,
:<math>
\dim U_{(k)} = \dim V_{(k)} - \dim V_{(k-2)} = \binom{n+k-1}{k}- \binom{n+k-3}{k}
</math>

=== Tensor products ===

In the stable range <math>|\mu|+|\nu|\leq \left[\frac{n}{2}\right]</math>, the tensor product multiplicities that appear in the tensor product decomposition <math>U_\lambda\otimes U_\mu = \oplus_\nu N_{\lambda,\mu,\nu} U_\nu</math> are [[Littlewood–Richardson rule#Newell-Littlewood numbers|Newell-Littlewood numbers]], which do not depend on <math>n</math>.<ref name="gao20"/> Beyond the stable range, the tensor product multiplicities become <math>n</math>-dependent modifications of the Newell-Littlewood numbers.<ref name="sam"/><ref name="gao20"/><ref name="Koike">{{CiteQ|Q56443390}}</ref> For example, for <math>n\geq 12</math>, we have 
:<math>
\begin{align} {}
 [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{align}
</math>

=== Branching rules from the general linear group ===

Since the orthogonal group is a subgroup of the general linear group, representations of <math>GL(n)</math> can be decomposed into representations of <math>O(n)</math>. The decomposition of a tensor representation is given in terms of [[Littlewood-Richardson coefficient]]s <math>c_{\lambda,\mu}^\nu</math> by the Littlewood restriction rule<ref name="htw03"/> 
:<math>
V_\nu^{GL(n)} = \sum_{\lambda,\mu} c_{\lambda,2\mu}^\nu U_\lambda^{O(n)}
</math>
where <math>2\mu</math> is a partition into even integers. The rule is valid in the stable range <math>2|\nu|,\tilde{\lambda}_1+\tilde{\lambda}_2\leq n </math>. The generalization to mixed tensor representations is 
:<math>
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)}
</math>
Similar branching rules can be written for the symplectic group.<ref name="htw03"/>

== [[Symplectic group]] ==

=== Representations ===

The finite-dimensional irreducible representations of the symplectic group <math>Sp(2n,\mathbb{C})</math> are parametrized by Young diagrams with at most <math>n</math> rows. The dimension of the corresponding representation is<ref name="Hamermesh"/>  
:<math>
\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}
</math>
There is also an expression as a factorized polynomial in <math>n</math>:<ref name="El Samra">{{cite Q|Q104601301}}</ref>
:<math>
\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)}
</math>

=== 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 ==

* [http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/form.html Lie online service], an online interface to the [http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/ Lie software].

== References ==
{{Reflist|refs=
<ref name="art24">{{cite arxiv | last=Artamonov | first=Dmitry | title=Calculation of <math>6j</math>-symbols for the Lie algebra <math>\mathfrak{gl}_n</math> | date=2024-05-09 | eprint=2405.05628 | class=math.RT}}</ref>
<ref name="htw03">{{cite journal|last1=Howe|first1=Roger| authorlink=Roger Evans Howe|last2=Tan|first2=Eng-Chye|last3= Willenbring|first3= Jeb F.|authorlink3=Jeb Willenbring|title=Stable branching rules for classical symmetric pairs|journal=Transactions of the American Mathematical Society|volume= 357|issue=4|year=2005|pages= 1601–1626|doi=10.1090/S0002-9947-04-03722-5|doi-access=free|arxiv=math/0311159}}</ref>
<ref name="haw13">{{cite arXiv | last=Hawkes | first=Graham | title=An Elementary Proof of the Hook Content Formula | date=2013-10-19 | class=math.CO |eprint=1310.5919v2}}</ref>
<ref name="sam">{{cite web | author = Steven Sam|title=Littlewood-Richardson coefficients for classical groups | website=Concrete Nonsense | date=2010-01-18 | url=https://concretenonsense.wordpress.com/2010/01/18/littlewood-richardson-coefficients-for-classical-groups/ | archive-url=https://web.archive.org/web/20190618084957/https://concretenonsense.wordpress.com/2010/01/18/littlewood-richardson-coefficients-for-classical-groups/ | archive-date=2019-06-18 | url-status=live | access-date=2021-01-05}}</ref>
<ref name="gao20">{{cite journal | last1=Gao | first1=Shiliang | last2=Orelowitz | first2=Gidon | last3=Yong | first3=Alexander | title=Newell-Littlewood numbers | journal=Transactions of the American Mathematical Society | year=2021 | volume=374 | issue=9 | pages=6331–6366 | doi=10.1090/tran/8375 | arxiv=2005.09012v1| s2cid=218684561 }}</ref>
<ref name="Hamermesh">{{cite book | last=Hamermesh | first=Morton | title=Group theory and its application to physical problems | publisher=Dover Publications | publication-place=New York | year=1989 | isbn=0-486-66181-4 | oclc=20218471}}</ref>
}}

[[Category:Representation theory of Lie groups]]
[[Category:Lie groups]]