Representations of classical Lie groups

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

[math]\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} }[/math]

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

The dimensions of these irreducible representations are^[1]

[math]\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)} }[/math]

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

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

Examples of tensor representations:

General irreducible representations

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

• [math]\displaystyle{ V_{\lambda}=V_{\lambda()}, V = V_{(1)()} }[/math]

• [math]\displaystyle{ (V_{\lambda\mu})^* = V_{\mu\lambda} }[/math]

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

• The dimension of [math]\displaystyle{ V_{\lambda\mu} }[/math] where [math]\displaystyle{ \lambda = (\lambda_1,\dots,\lambda_r), \mu=(\mu_1,\dots,\mu_s) }[/math] is

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

Tensor products

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

[math]\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}}, }[/math]

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

[math]\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}, }[/math]

where the natural integers [math]\displaystyle{ c_{\lambda,\mu}^\nu }[/math] 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 [math]\displaystyle{ O(n,\mathbb{C}) }[/math] and special orthogonal group [math]\displaystyle{ SO(n,\mathbb{C}) }[/math] have spin representations, which are projective representations of these groups, i.e. representations of their universal covering groups.

Construction of representations

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

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

[math]\displaystyle{ V_{(k)} = U_{(k)} \oplus V_{(k-2)} }[/math]

Case of the special orthogonal group

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

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

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

Dimensions

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

[math]\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} }[/math]

[math]\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} }[/math]

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

[math]\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)} }[/math]

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

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

[math]\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} }[/math]

Branching rules from the general linear group

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

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

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

[math]\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)} }[/math]

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

Symplectic group

Representations

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

[math]\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} }[/math]

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

[math]\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)} }[/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

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

References

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