# Representations of classical Lie groups

In mathematics, the finite-dimensional representations of the complex classical Lie groups $\displaystyle{ GL(n,\mathbb{C}) }$, $\displaystyle{ SL(n,\mathbb{C}) }$, $\displaystyle{ O(n,\mathbb{C}) }$, $\displaystyle{ SO(n,\mathbb{C}) }$, $\displaystyle{ Sp(2n,\mathbb{C}) }$, can be constructed using the general representation theory of semisimple Lie algebras. The groups $\displaystyle{ SL(n,\mathbb{C}) }$, $\displaystyle{ SO(n,\mathbb{C}) }$, $\displaystyle{ Sp(2n,\mathbb{C}) }$ are indeed simple Lie groups, and their finite-dimensional representations coincide[1] with those of their maximal compact subgroups, respectively $\displaystyle{ SU(n) }$, $\displaystyle{ SO(n) }$, $\displaystyle{ Sp(n) }$. In the classification of simple Lie algebras, the corresponding algebras are

\displaystyle{ \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} }

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 $\displaystyle{ V=\mathbb{C}^n }$ be the defining representation of the general linear group $\displaystyle{ GL(n,\mathbb{C}) }$. Tensor representations are the subrepresentations of $\displaystyle{ V^{\otimes k} }$ (these are sometimes called polynomial representations). The irreducible subrepresentations of $\displaystyle{ V^{\otimes k} }$ are the images of $\displaystyle{ V }$ by Schur functors $\displaystyle{ \mathbb{S}^\lambda }$ associated to partitions $\displaystyle{ \lambda }$ of $\displaystyle{ k }$ into at most $\displaystyle{ n }$ integers, i.e. to Young diagrams of size $\displaystyle{ \lambda_1+\cdots + \lambda_n = k }$ with $\displaystyle{ \lambda_{n+1}=0 }$. (If $\displaystyle{ \lambda_{n+1}\gt 0 }$ then $\displaystyle{ \mathbb{S}^\lambda(V)=0 }$.) Schur functors are defined using Young symmetrizers of the symmetric group $\displaystyle{ S_k }$, which acts naturally on $\displaystyle{ V^{\otimes k} }$. We write $\displaystyle{ V_\lambda = \mathbb{S}^\lambda(V) }$.

The dimensions of these irreducible representations are[1]

$\displaystyle{ \dim V_\lambda = \prod_{1\leq i \lt j \leq n}\frac{\lambda_i-\lambda_j +j-i}{j-i} = \prod_{(i,j)\in \lambda} \frac{n-i+j}{h_\lambda(i,j)} }$

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

• 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] $\displaystyle{ \chi_\lambda(g) = s_\lambda(x_1,\dots, x_n) }$ where $\displaystyle{ x_1,\dots ,x_n }$ are the eigenvalues of $\displaystyle{ g\in GL(n,\mathbb{C}) }$.
• The second formula for the dimension is sometimes called Stanley's hook content formula.[2]

Examples of tensor representations:

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

### General irreducible representations

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

• $\displaystyle{ V_{\lambda}=V_{\lambda()}, V = V_{(1)()} }$
• $\displaystyle{ (V_{\lambda\mu})^* = V_{\mu\lambda} }$
• For $\displaystyle{ k \in \mathbb Z }$, denoting $\displaystyle{ D_k }$ the one-dimensional representation in which $\displaystyle{ GL(n,\mathbb C) }$ acts by $\displaystyle{ (\det)^k }$, $\displaystyle{ V_{\lambda_1,\dots,\lambda_n} = V_{\lambda_1+k,\dots,\lambda_n+k} \otimes D_{-k} }$. If $\displaystyle{ k }$ is large enough that $\displaystyle{ \lambda_n + k \geq 0 }$, this gives an explicit description of $\displaystyle{ V_{\lambda_1, \dots,\lambda_n} }$ in terms of a Schur functor.
• The dimension of $\displaystyle{ V_{\lambda\mu} }$ where $\displaystyle{ \lambda = (\lambda_1,\dots,\lambda_r), \mu=(\mu_1,\dots,\mu_s) }$ is
$\displaystyle{ \dim(V_{\lambda\mu}) = d_\lambda d_\mu \prod_{i=1}^r \frac{(1-i-s+n)_{\lambda_i}}{(1-i+r)_{\lambda_i}} \prod_{j=1}^s \frac{(1-j-r+n)_{\mu_i}}{(1-j+s)_{\mu_i}}\prod_{i=1}^r \prod_{j=1}^s \frac{n+1 + \lambda_i + \mu_j - i- j }{n+1 -i -j } }$ where $\displaystyle{ d_\lambda = \prod_{1 \leq i \lt j \leq r} \frac{\lambda_i - \lambda_j + j - i}{j-i} }$.[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

Two representations $\displaystyle{ V_{\lambda},V_{\lambda'} }$ of $\displaystyle{ GL(n,\mathbb{C}) }$ are equivalent as representations of the special linear group $\displaystyle{ SL(n,\mathbb{C}) }$ if and only if there is $\displaystyle{ k\in\mathbb{Z} }$ such that $\displaystyle{ \forall i,\ \lambda_i-\lambda'_i=k }$.[1] For instance, the determinant representation $\displaystyle{ V_{(1^n)} }$ is trivial in $\displaystyle{ SL(n,\mathbb{C}) }$, i.e. it is equivalent to $\displaystyle{ V_{()} }$. In particular, irreducible representations of $\displaystyle{ SL(n,\mathbb C) }$ 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 $\displaystyle{ GL(n,\mathbb C) }$. The complexification of its Lie algebra $\displaystyle{ \mathfrak u(n) = \{a \in \mathcal M(n,\mathbb C), a^\dagger + a = 0\} }$ is the algebra $\displaystyle{ \mathfrak{gl}(n,\mathbb C) }$. In Lie theoretic terms, $\displaystyle{ U(n) }$ is the compact real form of $\displaystyle{ GL(n,\mathbb C) }$, 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 $\displaystyle{ U(n) \rightarrow GL(n,\mathbb C) }$. [5]

### Tensor products

Tensor products of finite-dimensional representations of $\displaystyle{ GL(n,\mathbb{C}) }$ are given by the following formula:[6]

$\displaystyle{ V_{\lambda_1\mu_1} \otimes V_{\lambda_2\mu_2} = \bigoplus_{\nu,\rho} V_{\nu\rho}^{\oplus \Gamma^{\nu\rho}_{\lambda_1\mu_1,\lambda_2\mu_2}}, }$

where $\displaystyle{ \Gamma^{\nu\rho}_{\lambda_1\mu_1,\lambda_2\mu_2} = 0 }$ unless $\displaystyle{ |\nu| \leq |\lambda_1| + |\lambda_2| }$ and $\displaystyle{ |\rho| \leq |\mu_1| + |\mu_2| }$. Calling $\displaystyle{ l(\lambda) }$ the number of lines in a tableau, if $\displaystyle{ l(\lambda_1) + l(\lambda_2) + l(\mu_1) + l(\mu_2) \leq n }$, then

$\displaystyle{ \Gamma^{\nu\rho}_{\lambda_1\mu_1,\lambda_2\mu_2} = \sum_{\alpha,\beta,\eta,\theta} \left(\sum_\kappa c^{\lambda_1}_{\kappa,\alpha} c^{\mu_2}_{\kappa,\beta}\right)\left(\sum_\gamma c^{\lambda_2}_{\gamma,\eta}c^{\mu_1}_{\gamma,\theta}\right)c^{\nu}_{\alpha,\theta}c^{\rho}_{\beta,\eta}, }$

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

Below are a few examples of such tensor products:

$\displaystyle{ R_1 }$ $\displaystyle{ R_2 }$ Tensor product $\displaystyle{ R_1 \otimes R_2 }$
$\displaystyle{ V_{\lambda()} }$ $\displaystyle{ V_{\mu()} }$ $\displaystyle{ \sum_\nu c^\nu_{\lambda \mu}V_{\nu()} }$
$\displaystyle{ V_{\lambda()} }$ $\displaystyle{ V_{()\mu} }$ $\displaystyle{ \sum_{\kappa,\nu,\rho} c^\lambda_{\kappa\nu} c^{\mu}_{\kappa\rho} V_{\nu\rho} }$
$\displaystyle{ V_{()(1)} }$ $\displaystyle{ V_{(1)()} }$ $\displaystyle{ V_{(1)(1)} + V_{()()} }$
$\displaystyle{ V_{()(1)} }$ $\displaystyle{ V_{(k)()} }$ $\displaystyle{ V_{(k)(1)} + V_{(k-1)()} }$
$\displaystyle{ V_{(1)()} }$ $\displaystyle{ V_{(k)()} }$ $\displaystyle{ V_{(k+1)()} + V_{(k,1)()} }$
$\displaystyle{ V_{(1)(1)} }$ $\displaystyle{ V_{(1)(1)} }$ $\displaystyle{ V_{(2)(2)} + V_{(2)(11)} + V_{(11)(2)} + V_{(11)(11)} + 2V_{(1)(1)} + V_{()()} }$

## Orthogonal group and special orthogonal group

In addition to the Lie group representations described here, the orthogonal group $\displaystyle{ O(n,\mathbb{C}) }$ and special orthogonal group $\displaystyle{ SO(n,\mathbb{C}) }$ have spin representations, which are projective representations of these groups, i.e. representations of their universal covering groups.

### Construction of representations

Since $\displaystyle{ O(n,\mathbb{C}) }$ is a subgroup of $\displaystyle{ GL(n,\mathbb{C}) }$, any irreducible representation of $\displaystyle{ GL(n,\mathbb{C}) }$ is also a representation of $\displaystyle{ O(n,\mathbb{C}) }$, which may however not be irreducible. In order for a tensor representation of $\displaystyle{ O(n,\mathbb{C}) }$ to be irreducible, the tensors must be traceless.[7]

Irreducible representations of $\displaystyle{ O(n,\mathbb{C}) }$ are parametrized by a subset of the Young diagrams associated to irreducible representations of $\displaystyle{ GL(n,\mathbb{C}) }$: the diagrams such that the sum of the lengths of the first two columns is at most $\displaystyle{ n }$.[7] The irreducible representation $\displaystyle{ U_\lambda }$ that corresponds to such a diagram is a subrepresentation of the corresponding $\displaystyle{ GL(n,\mathbb{C}) }$ representation $\displaystyle{ V_\lambda }$. For example, in the case of symmetric tensors,[1]

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

### Case of the special orthogonal group

The antisymmetric tensor $\displaystyle{ U_{(1^n)} }$ is a one-dimensional representation of $\displaystyle{ O(n,\mathbb{C}) }$, which is trivial for $\displaystyle{ SO(n,\mathbb{C}) }$. Then $\displaystyle{ U_{(1^n)}\otimes U_\lambda = U_{\lambda'} }$ where $\displaystyle{ \lambda' }$ is obtained from $\displaystyle{ \lambda }$ by acting on the length of the first column as $\displaystyle{ \tilde{\lambda}_1\to n-\tilde{\lambda}_1 }$.

• For $\displaystyle{ n }$ odd, the irreducible representations of $\displaystyle{ SO(n,\mathbb{C}) }$ are parametrized by Young diagrams with $\displaystyle{ \tilde{\lambda}_1\leq\frac{n-1}{2} }$ rows.
• For $\displaystyle{ n }$ even, $\displaystyle{ U_\lambda }$ is still irreducible as an $\displaystyle{ SO(n,\mathbb{C}) }$ representation if $\displaystyle{ \tilde{\lambda}_1\leq\frac{n}{2}-1 }$, but it reduces to a sum of two inequivalent $\displaystyle{ SO(n,\mathbb{C}) }$ representations if $\displaystyle{ \tilde{\lambda}_1=\frac{n}{2} }$.[7]

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

### Dimensions

The dimensions of irreducible representations of $\displaystyle{ SO(n,\mathbb{C}) }$ are given by a formula that depends on the parity of $\displaystyle{ n }$:[4]

$\displaystyle{ (n\text{ even}) \qquad \dim U_\lambda = \prod_{1\leq i\lt j\leq \frac{n}{2}} \frac{\lambda_i-\lambda_j-i+j}{-i+j}\cdot \frac{\lambda_i+\lambda_j+n-i-j}{n-i-j} }$
$\displaystyle{ (n\text{ odd}) \qquad \dim U_\lambda = \prod_{1\leq i\lt j\leq \frac{n-1}{2}} \frac{\lambda_i-\lambda_j-i+j}{-i+j} \prod_{1\leq i\leq j\leq \frac{n-1}{2}} \frac{\lambda_i+\lambda_j+n-i-j}{n-i-j} }$

There is also an expression as a factorized polynomial in $\displaystyle{ n }$:[4]

$\displaystyle{ \dim U_\lambda = \prod_{(i,j)\in \lambda,\ i\geq j} \frac{n+\lambda_i+\lambda_j-i-j}{h_\lambda(i,j)} \prod_{(i,j)\in \lambda,\ i\lt j} \frac{n-\tilde{\lambda}_i-\tilde{\lambda}_j+i+j-2}{h_\lambda(i,j)} }$

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

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

### Tensor products

In the stable range $\displaystyle{ |\mu|+|\nu|\leq \left[\frac{n}{2}\right] }$, the tensor product multiplicities that appear in the tensor product decomposition $\displaystyle{ U_\lambda\otimes U_\mu = \oplus_\nu N_{\lambda,\mu,\nu} U_\nu }$ are Newell-Littlewood numbers, which do not depend on $\displaystyle{ n }$.[8] Beyond the stable range, the tensor product multiplicities become $\displaystyle{ n }$-dependent modifications of the Newell-Littlewood numbers.[9][8][10] For example, for $\displaystyle{ n\geq 12 }$, we have

\displaystyle{ \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} }

### Branching rules from the general linear group

Since the orthogonal group is a subgroup of the general linear group, representations of $\displaystyle{ GL(n) }$ can be decomposed into representations of $\displaystyle{ O(n) }$. The decomposition of a tensor representation is given in terms of Littlewood-Richardson coefficients $\displaystyle{ c_{\lambda,\mu}^\nu }$ by the Littlewood restriction rule[11]

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

where $\displaystyle{ 2\mu }$ is a partition into even integers. The rule is valid in the stable range $\displaystyle{ 2|\nu|,\tilde{\lambda}_1+\tilde{\lambda}_2\leq n }$. The generalization to mixed tensor representations is

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

Similar branching rules can be written for the symplectic group.[11]

## Symplectic group

### Representations

The finite-dimensional irreducible representations of the symplectic group $\displaystyle{ Sp(2n,\mathbb{C}) }$ are parametrized by Young diagrams with at most $\displaystyle{ n }$ rows. The dimension of the corresponding representation is[7]

$\displaystyle{ \dim W_\lambda = \prod_{i=1}^n \frac{\lambda_i+n-i+1}{n-i+1} \prod_{1\leq i\lt j\leq n} \frac{\lambda_i-\lambda_j+j-i}{j-i} \cdot \frac{\lambda_i+\lambda_j+2n-i-j+2}{2n-i-j+2} }$

There is also an expression as a factorized polynomial in $\displaystyle{ n }$:[4]

$\displaystyle{ \dim W_\lambda = \prod_{(i,j)\in \lambda,\ i\gt j} \frac{n+\lambda_i+\lambda_j-i-j+2}{h_\lambda(i,j)} \prod_{(i,j)\in \lambda,\ i\leq j} \frac{n-\tilde{\lambda}_i-\tilde{\lambda}_j+i+j}{h_\lambda(i,j)} }$

### 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.