Representations Of Classical Lie Groups

In mathematics, the finite-dimensional representations of the complex classical Lie groups
$GL(n,\mathbb)$, $SL(n,\mathbb)$, $O(n,\mathbb)$, $SO(n,\mathbb)$, $Sp(2n,\mathbb)$,
can be constructed using the general representation theory of semisimple Lie algebras. The groups
$SL(n,\mathbb)$, $SO(n,\mathbb)$, $Sp(2n,\mathbb)$ 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
:$\begin SL(n,\mathbb)&\to A_ \\ SO(n_\text,\mathbb)&\to B_ \\ SO(n_\text,\mathbb) &\to D_ \\ Sp(2n,\mathbb)&\to C_n \end$
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 $V=\mathbb^n$ be the defining representation of the general linear group $GL(n,\mathbb)$. Tensor representations are the subrepresentations of $V^$ (these are sometimes called polynomial representations). The irreducible subrepresentations of $V^$ are the images of $V$ by Schur functors $\mathbb^\lambda$ 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 $\lambda_=0$. (If $\lambda_>0$ then $\mathbb^\lambda(V)=0$.) Schur functors are defined using Young symmetrizers of the symmetric group $S_k$, which acts naturally on $V^$. We write $V_\lambda = \mathbb^\lambda(V)$.

The dimensions of these irreducible representations are
:$\dim V_\lambda = \prod_\frac = \prod_ \frac$
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 $g\in GL(n,\mathbb)$.
* 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 $V^ \otimes (V^*)^$, 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 $\lambda_k \geq 0, \lambda_ \leq 0$, we can associate to $(\lambda_1, \dots ,\lambda_n)$ the pair of Young tableaux ( lambda_1\dots\lambda_k \lambda_n,\dots,-\lambda_ . This shows that irreducible representations of $GL(n,\mathbb C)$ can be labeled by pairs of Young tableaux . Let us denote $V_ = V_$ 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,

* $V_=V_, V = V_$

* $(V_)^* = V_$

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

* The dimension of $V_$ where $\lambda = (\lambda_1,\dots,\lambda_r), \mu=(\mu_1,\dots,\mu_s)$ is
:$\dim(V_) = d_\lambda d_\mu \prod_^r \frac \prod_^s \frac\prod_^r \prod_^s \frac$ where $d_\lambda = \prod_ \frac$. 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 $V_,V_$ of $GL(n,\mathbb)$ are equivalent as representations of the special linear group $SL(n,\mathbb)$ if and only if there is $k\in\mathbb$ such that $\forall i,\ \lambda_i-\lambda'_i=k$. For instance, the determinant representation $V_$ is trivial in $SL(n,\mathbb)$, 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) = \$ is the algebra $\mathfrak(n,\mathbb C)$. 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 $GL(n,\mathbb)$ are given by the following formula:

:$V_ \otimes V_ = \bigoplus_ V_^,$
where $\Gamma^_ = 0$ 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
:$\Gamma^_ = \sum_ \left(\sum_\kappa c^_ c^_\right)\left(\sum_\gamma c^_c^_\right)c^_c^_,$
where the natural integers $c_^\nu$ 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.

Orthogonal group and special orthogonal group

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

Construction of representations

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

Irreducible representations of $O(n,\mathbb)$ are parametrized by a subset of the Young diagrams associated to irreducible representations of $GL(n,\mathbb)$: 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 $GL(n,\mathbb)$ representation $V_\lambda$. For example, in the case of symmetric tensors,
:$V_ = U_ \oplus V_$

Case of the special orthogonal group

The antisymmetric tensor $U_$ is a one-dimensional representation of $O(n,\mathbb)$, which is trivial for $SO(n,\mathbb)$. Then $U_\otimes U_\lambda = U_$ where $\lambda'$ is obtained from $\lambda$ by acting on the length of the first column as $\tilde_1\to n-\tilde_1$.
* For $n$ odd, the irreducible representations of $SO(n,\mathbb)$ are parametrized by Young diagrams with $\tilde_1\leq\frac$ rows.
* For $n$ even, $U_\lambda$ is still irreducible as an $SO(n,\mathbb)$ representation if $\tilde_1\leq\frac-1$, but it reduces to a sum of two inequivalent $SO(n,\mathbb)$ representations if $\tilde_1=\frac$.

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

Dimensions

The dimensions of irreducible representations of $SO(n,\mathbb)$ are given by a formula that depends on the parity of $n$:
:$(n\text) \qquad \dim U_\lambda = \prod_ \frac\cdot \frac$
:$(n\text) \qquad \dim U_\lambda = \prod_ \frac \prod_ \frac$
There is also an expression as a factorized polynomial in $n$:
:$\dim U_\lambda = \prod_ \frac \prod_ \frac$
where $\lambda_i,\tilde_i,h_\lambda(i,j)$ are respectively row lengths, column lengths and hook lengths. In particular, antisymmetric representations have the same dimensions as their $GL(n,\mathbb)$ counterparts, $\dim U_=\dim V_$, but symmetric representations do not,
:$\dim U_ = \dim V_ - \dim V_ = \binom- \binom$

Tensor products

In the stable range , \mu, +, \nu, \leq \left frac\right/math>, the tensor product multiplicities that appear in the tensor product decomposition $U_\lambda\otimes U_\mu = \oplus_\nu N_ U_\nu$ 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
otimes &= + 1+ []
\\
otimes &= 1+ + [1]
\\
otimes 1&= 11+ 1+
\\
otimes 1&= 1 2 11 + 1
\\
\otimes &= 1
\\
otimes &= 1 2 1[]
\\
otimes 1&= 1 11+ 1
\\
1otimes 1&= 111+ 11+ 2+ + 1+ []
\\
[21]\otimes &=[321]+[411]+[42]+[51]+ 11 22 1 1 \end

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 $c_^\nu$ by the Littlewood restriction rule
:$V_\nu^ = \sum_ c_^\nu U_\lambda^$
where $2\mu$ is a partition into even integers. The rule is valid in the stable range $2, \nu, ,\tilde_1+\tilde_2\leq n$. The generalization to mixed tensor representations is
:$V_^ = \sum_ c_^\lambda c_^\mu c_^\nu U_\nu^$
Similar branching rules can be written for the symplectic group.

Symplectic group

Representations

The finite-dimensional irreducible representations of the symplectic group $Sp(2n,\mathbb)$ are parametrized by Young diagrams with at most $n$ rows. The dimension of the corresponding representation is
:$\dim W_\lambda = \prod_^n \frac \prod_ \frac \cdot \frac$
There is also an expression as a factorized polynomial in $n$:
:$\dim W_\lambda = \prod_ \frac \prod_ \frac$

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

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Lie software

References

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Representation theory of Lie groups
Lie groups