Representations of classical Lie groups

In mathematics, the finite-dimensional representations of the complex classical Lie groups $GL(n,\mathbb {C} )$, $SL(n,\mathbb {C} )$, $O(n,\mathbb {C} )$, $SO(n,\mathbb {C} )$, $Sp(2n,\mathbb {C} )$, can be constructed using the general representation theory of semisimple Lie algebras. The groups $SL(n,\mathbb {C} )$, $SO(n,\mathbb {C} )$, $Sp(2n,\mathbb {C} )$ are indeed simple Lie groups, and their finite-dimensional representations coincide with those of their maximal compact subgroups, respectively $SU(n)$, $SO(n)$, $Sp(n)$. In the classification of simple Lie algebras, the corresponding algebras are

${\begin{aligned}SL(n,\mathbb {C} )&\to A_{n-1}\\SO(n_{\text{odd}},\mathbb {C} )&\to B_{\frac {n-1}{2}}\\SO(n_{\text{even}},\mathbb {C} )&\to D_{\frac {n}{2}}\\Sp(2n,\mathbb {C} )&\to C_{n}\end{aligned}}$

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 $V=\mathbb {C} ^{n}$ be the defining representation of the general linear group $GL(n,\mathbb {C} )$. Tensor representations are the subrepresentations of $V^{\otimes k}$ (these are sometimes called polynomial representations). The irreducible subrepresentations of $V^{\otimes k}$ are the images of $V$ by Schur functors $\mathbb {S} ^{\lambda }$ associated to integer partitions $\lambda$ of $k$ into at most $n$ integers, i.e. to Young diagrams of size $\lambda _{1}+\cdots +\lambda _{n}=k$ with $\lambda _{n+1}=0$. (If $\lambda _{n+1}>0$ then $\mathbb {S} ^{\lambda }(V)=0$.) Schur functors are defined using Young symmetrizers of the symmetric group $S_{k}$, which acts naturally on $V^{\otimes k}$. We write $V_{\lambda }=\mathbb {S} ^{\lambda }(V)$.

The dimensions of these irreducible representations are

$\dim V_{\lambda }=\prod _{1\leq i<j\leq n}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}=\prod _{(i,j)\in \lambda }{\frac {n-i+j}{h_{\lambda }(i,j)}}$

where $h_{\lambda }(i,j)$ is the hook length of the cell $(i,j)$ in the Young diagram $\lambda$.

• The first formula for the dimension is a special case of a formula that gives the characters of representations in terms of Schur polynomials, $\chi _{\lambda }(g)=s_{\lambda }(x_{1},\dots ,x_{n})$ where $x_{1},\dots ,x_{n}$ are the eigenvalues of $g\in GL(n,\mathbb {C} )$.
• The second formula for the dimension is sometimes called Stanley's hook content formula.

Examples of tensor representations:

Tensor representation of $GL(n,\mathbb {C} )$	Dimension	Young diagram
Trivial representation	$1$	$()$
Determinant representation	$1$	$(1^{n})$
Defining representation $V$	$n$	$(1)$
Symmetric representation ${\text{Sym}}^{k}V$	${\binom {n+k-1}{k}}$	$(k)$
Antisymmetric representation $\Lambda ^{k}V$	${\binom {n}{k}}$	$(1^{k})$

General irreducible representations

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

• $V_{\lambda }=V_{\lambda ()},V=V_{(1)()}$

• $(V_{\lambda \mu })^{*}=V_{\mu \lambda }$

• For $k\in \mathbb {Z}$, denoting $D_{k}$ the one-dimensional representation in which $GL(n,\mathbb {C} )$ acts by $(\det )^{k}$, $V_{\lambda _{1},\dots ,\lambda _{n}}=V_{\lambda _{1}+k,\dots ,\lambda _{n}+k}\otimes D_{-k}$. If $k$ is large enough that $\lambda _{n}+k\geq 0$, this gives an explicit description of $V_{\lambda _{1},\dots ,\lambda _{n}}$ in terms of a Schur functor.

• The dimension of $V_{\lambda \mu }$ where $\lambda =(\lambda _{1},\dots ,\lambda _{r}),\mu =(\mu _{1},\dots ,\mu _{s})$ is

$\dim(V_{\lambda \mu })=d_{\lambda }d_{\mu }\prod _{i=1}^{r}{\frac {(1-i-s+n)_{\lambda _{i}}}{(1-i+r)_{\lambda _{i}}}}\prod _{j=1}^{s}{\frac {(1-j-r+n)_{\mu _{i}}}{(1-j+s)_{\mu _{i}}}}\prod _{i=1}^{r}\prod _{j=1}^{s}{\frac {n+1+\lambda _{i}+\mu _{j}-i-j}{n+1-i-j}}$ where $d_{\lambda }=\prod _{1\leq i<j\leq r}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}$. 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 $V_{\lambda },V_{\lambda '}$ of $GL(n,\mathbb {C} )$ are equivalent as representations of the special linear group $SL(n,\mathbb {C} )$ if and only if there is $k\in \mathbb {Z}$ such that $\forall i,\ \lambda _{i}-\lambda '_{i}=k$. For instance, the determinant representation $V_{(1^{n})}$ is trivial in $SL(n,\mathbb {C} )$, i.e. it is equivalent to $V_{()}$. In particular, irreducible representations of $SL(n,\mathbb {C} )$ 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 $GL(n,\mathbb {C} )$. The complexification of its Lie algebra ${\mathfrak {u}}(n)=\{a\in {\mathcal {M}}(n,\mathbb {C} ),a^{\dagger }+a=0\}$ is the algebra ${\mathfrak {gl}}(n,\mathbb {C} )$. In Lie theoretic terms, $U(n)$ is the compact real form of $GL(n,\mathbb {C} )$, 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 $U(n)\rightarrow GL(n,\mathbb {C} )$.

Tensor products

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

$V_{\lambda _{1}\mu _{1}}\otimes V_{\lambda _{2}\mu _{2}}=\bigoplus _{\nu ,\rho }V_{\nu \rho }^{\oplus \Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }},$

where $\Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }=0$ unless $|\nu |\leq |\lambda _{1}|+|\lambda _{2}|$ and $|\rho |\leq |\mu _{1}|+|\mu _{2}|$. Calling $l(\lambda )$ the number of lines in a tableau, if $l(\lambda _{1})+l(\lambda _{2})+l(\mu _{1})+l(\mu _{2})\leq n$, then

$\Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }=\sum _{\alpha ,\beta ,\eta ,\theta }\left(\sum _{\kappa }c_{\kappa ,\alpha }^{\lambda _{1}}c_{\kappa ,\beta }^{\mu _{2}}\right)\left(\sum _{\gamma }c_{\gamma ,\eta }^{\lambda _{2}}c_{\gamma ,\theta }^{\mu _{1}}\right)c_{\alpha ,\theta }^{\nu }c_{\beta ,\eta }^{\rho },$

where the natural integers $c_{\lambda ,\mu }^{\nu }$ are Littlewood-Richardson coefficients.

Below are a few examples of such tensor products:

$R_{1}$	$R_{2}$	Tensor product $R_{1}\otimes R_{2}$
$V_{\lambda ()}$	$V_{\mu ()}$	$\sum _{\nu }c_{\lambda \mu }^{\nu }V_{\nu ()}$
$V_{\lambda ()}$	$V_{()\mu }$	$\sum _{\kappa ,\nu ,\rho }c_{\kappa \nu }^{\lambda }c_{\kappa \rho }^{\mu }V_{\nu \rho }$
$V_{()(1)}$	$V_{(1)()}$	$V_{(1)(1)}+V_{()()}$
$V_{()(1)}$	$V_{(k)()}$	$V_{(k)(1)}+V_{(k-1)()}$
$V_{(1)()}$	$V_{(k)()}$	$V_{(k+1)()}+V_{(k,1)()}$
$V_{(1)(1)}$	$V_{(1)(1)}$	$V_{(2)(2)}+V_{(2)(11)}+V_{(11)(2)}+V_{(11)(11)}+2V_{(1)(1)}+V_{()()}$

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 $O(n,\mathbb {C} )$ and special orthogonal group $SO(n,\mathbb {C} )$ have spin representations, which are projective representations of these groups, i.e. representations of their universal covering groups.

Construction of representations

Since $O(n,\mathbb {C} )$ is a subgroup of $GL(n,\mathbb {C} )$, any irreducible representation of $GL(n,\mathbb {C} )$ is also a representation of $O(n,\mathbb {C} )$, which may however not be irreducible. In order for a tensor representation of $O(n,\mathbb {C} )$ to be irreducible, the tensors must be traceless.

Irreducible representations of $O(n,\mathbb {C} )$ are parametrized by a subset of the Young diagrams associated to irreducible representations of $GL(n,\mathbb {C} )$: the diagrams such that the sum of the lengths of the first two columns is at most $n$. The irreducible representation $U_{\lambda }$ that corresponds to such a diagram is a subrepresentation of the corresponding $GL(n,\mathbb {C} )$ representation $V_{\lambda }$. For example, in the case of symmetric tensors,

$V_{(k)}=U_{(k)}\oplus V_{(k-2)}$

Case of the special orthogonal group

The antisymmetric tensor $U_{(1^{n})}$ is a one-dimensional representation of $O(n,\mathbb {C} )$, which is trivial for $SO(n,\mathbb {C} )$. Then $U_{(1^{n})}\otimes U_{\lambda }=U_{\lambda '}$ where $\lambda '$ is obtained from $\lambda$ by acting on the length of the first column as ${\tilde {\lambda }}_{1}\to n-{\tilde {\lambda }}_{1}$.

• For $n$ odd, the irreducible representations of $SO(n,\mathbb {C} )$ are parametrized by Young diagrams with ${\tilde {\lambda }}_{1}\leq {\frac {n-1}{2}}$ rows.
• For $n$ even, $U_{\lambda }$ is still irreducible as an $SO(n,\mathbb {C} )$ representation if ${\tilde {\lambda }}_{1}\leq {\frac {n}{2}}-1$, but it reduces to a sum of two inequivalent $SO(n,\mathbb {C} )$ representations if ${\tilde {\lambda }}_{1}={\frac {n}{2}}$.

For example, the irreducible representations of $O(3,\mathbb {C} )$ correspond to Young diagrams of the types $(k\geq 0),(k\geq 1,1),(1,1,1)$. The irreducible representations of $SO(3,\mathbb {C} )$ correspond to $(k\geq 0)$, and $\dim U_{(k)}=2k+1$. On the other hand, the dimensions of the spin representations of $SO(3,\mathbb {C} )$ are even integers.

Dimensions

The dimensions of irreducible representations of $SO(n,\mathbb {C} )$ are given by a formula that depends on the parity of $n$:

$(n{\text{ even}})\qquad \dim U_{\lambda }=\prod _{1\leq i<j\leq {\frac {n}{2}}}{\frac {\lambda _{i}-\lambda _{j}-i+j}{-i+j}}\cdot {\frac {\lambda _{i}+\lambda _{j}+n-i-j}{n-i-j}}$

$(n{\text{ odd}})\qquad \dim U_{\lambda }=\prod _{1\leq i<j\leq {\frac {n-1}{2}}}{\frac {\lambda _{i}-\lambda _{j}-i+j}{-i+j}}\prod _{1\leq i\leq j\leq {\frac {n-1}{2}}}{\frac {\lambda _{i}+\lambda _{j}+n-i-j}{n-i-j}}$

There is also an expression as a factorized polynomial in $n$:

$\dim U_{\lambda }=\prod _{(i,j)\in \lambda ,\ i\geq j}{\frac {n+\lambda _{i}+\lambda _{j}-i-j}{h_{\lambda }(i,j)}}\prod _{(i,j)\in \lambda ,\ i<j}{\frac {n-{\tilde {\lambda }}_{i}-{\tilde {\lambda }}_{j}+i+j-2}{h_{\lambda }(i,j)}}$

where $\lambda _{i},{\tilde {\lambda }}_{i},h_{\lambda }(i,j)$ are respectively row lengths, column lengths and hook lengths. In particular, antisymmetric representations have the same dimensions as their $GL(n,\mathbb {C} )$ counterparts, $\dim U_{(1^{k})}=\dim V_{(1^{k})}$, but symmetric representations do not,

$\dim U_{(k)}=\dim V_{(k)}-\dim V_{(k-2)}={\binom {n+k-1}{k}}-{\binom {n+k-3}{k}}$

Tensor products

In the stable range $|\mu |+|\nu |\leq \left[{\frac {n}{2}}\right]$, the tensor product multiplicities that appear in the tensor product decomposition $U_{\lambda }\otimes U_{\mu }=\oplus _{\nu }N_{\lambda ,\mu ,\nu }U_{\nu }$ are Newell-Littlewood numbers, which do not depend on $n$. Beyond the stable range, the tensor product multiplicities become $n$-dependent modifications of the Newell-Littlewood numbers. For example, for $n\geq 12$, we have

${\begin{aligned}{}[1]\otimes [1]&=[2]+[11]+[]\\{}[1]\otimes [2]&=[21]+[3]+[1]\\{}[1]\otimes [11]&=[111]+[21]+[1]\\{}[1]\otimes [21]&=[31]+[22]+[211]+[2]+[11]\\{}[1]\otimes [3]&=[4]+[31]+[2]\\{}[2]\otimes [2]&=[4]+[31]+[22]+[2]+[11]+[]\\{}[2]\otimes [11]&=[31]+[211]+[2]+[11]\\{}[11]\otimes [11]&=[1111]+[211]+[22]+[2]+[11]+[]\\{}[21]\otimes [3]&=[321]+[411]+[42]+[51]+[211]+[22]+2[31]+[4]+[11]+[2]\end{aligned}}$

Branching rules from the general linear group

Since the orthogonal group is a subgroup of the general linear group, representations of $GL(n)$ can be decomposed into representations of $O(n)$. The decomposition of a tensor representation is given in terms of Littlewood-Richardson coefficients $c_{\lambda ,\mu }^{\nu }$ by the Littlewood restriction rule

$V_{\nu }^{GL(n)}=\sum _{\lambda ,\mu }c_{\lambda ,2\mu }^{\nu }U_{\lambda }^{O(n)}$

where $2\mu$ is a partition into even integers. The rule is valid in the stable range $2|\nu |,{\tilde {\lambda }}_{1}+{\tilde {\lambda }}_{2}\leq n$. The generalization to mixed tensor representations is

$V_{\lambda \mu }^{GL(n)}=\sum _{\alpha ,\beta ,\gamma ,\delta }c_{\alpha ,2\gamma }^{\lambda }c_{\beta ,2\delta }^{\mu }c_{\alpha ,\beta }^{\nu }U_{\nu }^{O(n)}$

Similar branching rules can be written for the symplectic group.

Symplectic group

Representations

The finite-dimensional irreducible representations of the symplectic group $Sp(2n,\mathbb {C} )$ are parametrized by Young diagrams with at most $n$ rows. The dimension of the corresponding representation is

$\dim W_{\lambda }=\prod _{i=1}^{n}{\frac {\lambda _{i}+n-i+1}{n-i+1}}\prod _{1\leq i<j\leq n}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}\cdot {\frac {\lambda _{i}+\lambda _{j}+2n-i-j+2}{2n-i-j+2}}$

There is also an expression as a factorized polynomial in $n$:

$\dim W_{\lambda }=\prod _{(i,j)\in \lambda ,\ i>j}{\frac {n+\lambda _{i}+\lambda _{j}-i-j+2}{h_{\lambda }(i,j)}}\prod _{(i,j)\in \lambda ,\ i\leq j}{\frac {n-{\tilde {\lambda }}_{i}-{\tilde {\lambda }}_{j}+i+j}{h_{\lambda }(i,j)}}$

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

• Lie online service, an online interface to the Lie software.