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{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.

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$$. \chi_\lambda(g) = s_\lambda(x_1,\dots, x_n) $$ where $$x_1,\dots ,x_n$$ are the eigenvalues of $$g\in GL(n,\mathbb{C})$$.
 * The first formula for the dimension is a special case of a formula that gives the characters of representations in terms of Schur polynomials, $$
 * The second formula for the dimension is sometimes called Stanley's hook content formula.

Examples of tensor representations:

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^{\nu\rho}_{\lambda_1\mu_1,\lambda_2\mu_2}}, $$ where $$ \Gamma^{\nu\rho}_{\lambda_1\mu_1,\lambda_2\mu_2} = 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^{\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}, $$ where the natural integers $$c_{\lambda,\mu}^\nu$$ are Littlewood-Richardson coefficients.

Below are a few examples of such tensor products:

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{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 $$GL(n)$$ 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.

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} \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.