Representations of classical groups

\begin{matrix} S L (n, C) & \to A_{n - 1} \\ S O (n_{odd}, C) & \to B_{\frac{n - 1}{2}} \\ S O (n_{even}, C) & \to D_{\frac{n}{2}} \\ S p (2 n, C) & \to C_{n} \end{matrix}

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 $V = C^{n}$ be the defining representation of the general linear group $G L (n, C)$ . 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 $S^{λ}$ associated to integer partitions $λ$ of $k$ into at most $n$ integers, i.e. to Young diagrams of size $λ_{1} + \dots + λ_{n} = k$ with $λ_{n + 1} = 0$ . (If $λ_{n + 1} > 0$ then $S^{λ} (V) = 0$ .) Schur functors are defined using Young symmetrizers of the symmetric group $S_{k}$ , which acts naturally on $V^{\otimes k}$ . We write $V_{λ} = S^{λ} (V)$ .

The dimensions of these irreducible representations are

\dim V_{λ} = \prod_{1 \leq i < j \leq n} \frac{λ_{i} - λ_{j} + j - i}{j - i} = \prod_{(i, j) \in λ} \frac{n - i + j}{h_{λ} (i, j)}

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

The first formula for the dimension is a special case of a formula that gives the characters of representations in terms of Schur polynomials, $χ_{λ} (g) = s_{λ} (x_{1}, \dots, x_{n})$ where $x_{1}, \dots, x_{n}$ are the eigenvalues of $g \in G L (n, C)$ .
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 $G L (n, C)$ are tensor representations. In general, irreducible representations of $G L (n, C)$ 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 $G L (n, C)$ is labeled by non increasing sequences of $n$ integers $λ_{1} \geq \dots \geq λ_{n}$ . If $λ_{k} \geq 0, λ_{k + 1} \leq 0$ , we can associate to $(λ_{1}, \dots, λ_{n})$ the pair of Young tableaux $([λ_{1} \dots λ_{k}], [- λ_{n}, \dots, - λ_{k + 1}])$ . This shows that irreducible representations of $G L (n, C)$ can be labeled by pairs of Young tableaux . Let us denote $V_{λ μ} = V_{λ_{1}, \dots, λ_{n}}$ the irreducible representation of $G L (n, C)$ corresponding to the pair $(λ, μ)$ or equivalently to the sequence $(λ_{1}, \dots, λ_{n})$ . With these notations,

$V_{λ} = V_{λ ()}, V = V_{(1) ()}$

$(V_{λ μ})^{*} = V_{μ λ}$

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

The dimension of $V_{λ μ}$ where $λ = (λ_{1}, \dots, λ_{r}), μ = (μ_{1}, \dots, μ_{s})$ is

\dim (V_{λ μ}) = d_{λ} d_{μ} \prod_{i = 1}^{r} \frac{(1 - i - s + n)_{λ_{i}}}{(1 - i + r)_{λ_{i}}} \prod_{j = 1}^{s} \frac{(1 - j - r + n)_{μ_{i}}}{(1 - j + s)_{μ_{i}}} \prod_{i = 1}^{r} \prod_{j = 1}^{s} \frac{n + 1 + λ_{i} + μ_{j} - i - j}{n + 1 - i - j}

where

d_{λ} = \prod_{1 \leq i < j \leq r} \frac{λ_{i} - λ_{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 $V_{λ}, V_{λ^{'}}$ of $G L (n, C)$ are equivalent as representations of the special linear group $S L (n, C)$ if and only if there is $k \in Z$ such that $\forall i, λ_{i} - λ_{i}^{'} = k$ . For instance, the determinant representation $V_{(1^{n})}$ is trivial in $S L (n, C)$ , i.e. it is equivalent to $V_{()}$ . In particular, irreducible representations of $S L (n, C)$ 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 $G L (n, C)$ . The complexification of its Lie algebra $u (n) = {a \in M (n, C), a^{†} + a = 0}$ is the algebra $g l (n, C)$ . In Lie theoretic terms, $U (n)$ is the compact real form of $G L (n, C)$ , 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 $U (n) \to G L (n, C)$ .

Tensor products

Tensor products of finite-dimensional representations of $G L (n, C)$ are given by the following formula:

V_{λ_{1} μ_{1}} \otimes V_{λ_{2} μ_{2}} = ⨁_{ν, ρ} V_{ν ρ}^{\oplus Γ_{λ_{1} μ_{1}, λ_{2} μ_{2}}^{ν ρ}},

where $Γ_{λ_{1} μ_{1}, λ_{2} μ_{2}}^{ν ρ} = 0$ unless $| ν | \leq | λ_{1} | + | λ_{2} |$ and $| ρ | \leq | μ_{1} | + | μ_{2} |$ . Calling $l (λ)$ the number of lines in a tableau, if $l (λ_{1}) + l (λ_{2}) + l (μ_{1}) + l (μ_{2}) \leq n$ , then

Γ_{λ_{1} μ_{1}, λ_{2} μ_{2}}^{ν ρ} = \sum_{α, β, η, θ} (\sum_{κ} c_{κ, α}^{λ_{1}} c_{κ, β}^{μ_{2}}) (\sum_{γ} c_{γ, η}^{λ_{2}} c_{γ, θ}^{μ_{1}}) c_{α, θ}^{ν} c_{β, η}^{ρ},

where the natural integers $c_{λ, μ}^{ν}$ are Littlewood-Richardson coefficients.

Below are a few examples of such tensor products:

In the case of tensor representations, 3-j symbols and 6-j symbols are known.

2 more chapters are free with a Lably account.

Orthogonal group and special orthogonal group

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

Construction of representations

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

Irreducible representations of $O (n, C)$ are parametrized by a subset of the Young diagrams associated to irreducible representations of $G L (n, C)$ : the diagrams such that the sum of the lengths of the first two columns is at most $n$ . The irreducible representation $U_{λ}$ that corresponds to such a diagram is a subrepresentation of the corresponding $G L (n, C)$ representation $V_{λ}$ . 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 $O (n, C)$ , which is trivial for $S O (n, C)$ . Then $U_{(1^{n})} \otimes U_{λ} = U_{λ^{'}}$ where $λ^{'}$ is obtained from $λ$ by acting on the length of the first column as ${\tilde{λ}}_{1} \to n - {\tilde{λ}}_{1}$ .

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

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

Dimensions

The dimensions of irreducible representations of $S O (n, C)$ are given by a formula that depends on the parity of $n$ :

(n even) \dim U_{λ} = \prod_{1 \leq i < j \leq \frac{n}{2}} \frac{λ_{i} - λ_{j} - i + j}{- i + j} \cdot \frac{λ_{i} + λ_{j} + n - i - j}{n - i - j}

(n odd) \dim U_{λ} = \prod_{1 \leq i < j \leq \frac{n - 1}{2}} \frac{λ_{i} - λ_{j} - i + j}{- i + j} \prod_{1 \leq i \leq j \leq \frac{n - 1}{2}} \frac{λ_{i} + λ_{j} + n - i - j}{n - i - j}

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

\dim U_{λ} = \prod_{(i, j) \in λ, i \geq j} \frac{n + λ_{i} + λ_{j} - i - j}{h_{λ} (i, j)} \prod_{(i, j) \in λ, i < j} \frac{n - {\tilde{λ}}_{i} - {\tilde{λ}}_{j} + i + j - 2}{h_{λ} (i, j)}

where $λ_{i}, {\tilde{λ}}_{i}, h_{λ} (i, j)$ are respectively row lengths, column lengths and hook lengths. In particular, antisymmetric representations have the same dimensions as their $G L (n, C)$ 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 $| μ | + | ν | \leq [\frac{n}{2}]$ , the tensor product multiplicities that appear in the tensor product decomposition $U_{λ} \otimes U_{μ} = \oplus_{ν} N_{λ, μ, ν} U_{ν}$ 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{matrix} [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{matrix}

Branching rules from the general linear group

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

V_{ν}^{G L (n)} = \sum_{λ, μ} c_{λ, 2 μ}^{ν} U_{λ}^{O (n)}

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

V_{λ μ}^{G L (n)} = \sum_{α, β, γ, δ} c_{α, 2 γ}^{λ} c_{β, 2 δ}^{μ} c_{α, β}^{ν} U_{ν}^{O (n)}

Similar branching rules can be written for the symplectic group.

Symplectic group