Representations of classical Lie groups

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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

${\displaystyle \begin{array}{rl}SL(n,\mathbb{C})& \to A_{n-1}\\ SO(n_{\text{odd}},\mathbb{C})& \to B_{\frac{n-1}{2}}\\ SO(n_{\text{even}},\mathbb{C})& \to D_{\frac{n}{2}}\\ Sp(2n,\mathbb{C})& \to C_n\end{array}}$

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 ${\displaystyle {\lambda }_{n+1}>0}$ 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]

${\displaystyle \dim V_{\lambda }=\prod _{1\le i<j\le n}\frac{{\lambda }_i-{\lambda }_j+j-i}{j-i}=\prod _{(i,j)\in \lambda }\frac{n-i+j}{h_{\lambda }(i,j)}}$

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:

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 ${\displaystyle V^{\otimes r}\otimes (V^{\ast }{)}^{\otimes s}}$, where ${\displaystyle V^{\ast }}$ 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 ${\displaystyle {\lambda }_1\ge \cdots \ge {\lambda }_n}$. If ${\displaystyle {\lambda }_k\ge 0,{\lambda }_{k+1}\le 0}$, 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,

• ${\displaystyle V_{\lambda }=V_{\lambda ()},V=V_{(1)()}}$

• ${\displaystyle (V_{\lambda \mu }{)}^{\ast }=V_{\mu \lambda }}$

• 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 ${\displaystyle {\lambda }_n+k\ge 0}$, 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 ${\displaystyle d_{\lambda }=\prod _{1\le i<j\le r}\frac{{\lambda }_i-{\lambda }_j+j-i}{j-i}}$.^[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 ${\displaystyle V_{\lambda },V_{{\lambda }^{\prime }}}$ 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 ${\displaystyle \mathrm{\forall }i,{\lambda }_i-{\lambda }_i^{\prime }=k}$.^[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 ${\displaystyle \mathfrak{u}(n)=\{a\in \mathcal{M}(n,\mathbb{C}),a^{†}+a=0\}}$ 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 ${\displaystyle U(n)\to GL(n,\mathbb{C})}$. ^[5]

Tensor products

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

${\displaystyle V_{{\lambda }_1{\mu }_1}\otimes V_{{\lambda }_2{\mu }_2}=\underset{\nu ,\rho }{⨁}{V}_{\nu \rho }^{\oplus {\mathrm{\Gamma }}_{{\lambda }_1{\mu }_1,{\lambda }_2{\mu }_2}^{\nu \rho }},}$

where ${\displaystyle {\mathrm{\Gamma }}_{{\lambda }_1{\mu }_1,{\lambda }_2{\mu }_2}^{\nu \rho }=0}$ unless ${\displaystyle |\nu |\le |{\lambda }_1|+|{\lambda }_2|}$ and ${\displaystyle |\rho |\le |{\mu }_1|+|{\mu }_2|}$. Calling ${\displaystyle l(\lambda )}$ the number of lines in a tableau, if ${\displaystyle l({\lambda }_1)+l({\lambda }_2)+l({\mu }_1)+l({\mu }_2)\le n}$, then

${\displaystyle {\mathrm{\Gamma }}_{{\lambda }_1{\mu }_1,{\lambda }_2{\mu }_2}^{\nu \rho }=\sum _{\alpha ,\beta ,\eta ,\theta }\left(\sum _{\kappa }{c}_{\kappa ,\alpha }^{{\lambda }_1}{c}_{\kappa ,\beta }^{{\mu }_2}\right)\left(\sum _{\gamma }{c}_{\gamma ,\eta }^{{\lambda }_2}{c}_{\gamma ,\theta }^{{\mu }_1}\right)c_{\alpha ,\theta }^{\nu }{c}_{\beta ,\eta }^{\rho },}$

where the natural integers ${\displaystyle c_{\lambda ,\mu }^{\nu }}$ 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 ${\displaystyle O(n,\mathbb{C})}$ 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 ${\displaystyle U_{(1^n)}\otimes U_{\lambda }=U_{{\lambda }^{\prime }}}$ where ${\displaystyle {\lambda }^{\prime }}$ 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 ${\displaystyle {\tilde{\lambda }}_1\le \frac{n-1}{2}}$ rows.
• For ${\displaystyle n}$ even, ${\displaystyle U_{\lambda }}$ is still irreducible as an ${\displaystyle SO(n,\mathbb{C})}$ representation if ${\displaystyle {\tilde{\lambda }}_1\le \frac{n}{2}-1}$, 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 ${\displaystyle (k\ge 0),(k\ge 1,1),(1,1,1)}$. The irreducible representations of ${\displaystyle SO(3,\mathbb{C})}$ correspond to ${\displaystyle (k\ge 0)}$, 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]

${\displaystyle (n\text{ even})\dim U_{\lambda }=\prod _{1\le i<j\le \frac{n}{2}}\frac{{\lambda }_i-{\lambda }_j-i+j}{-i+j}\cdot \frac{{\lambda }_i+{\lambda }_j+n-i-j}{n-i-j}}$

${\displaystyle (n\text{ odd})\dim U_{\lambda }=\prod _{1\le i<j\le \frac{n-1}{2}}\frac{{\lambda }_i-{\lambda }_j-i+j}{-i+j}\prod _{1\le i\le j\le \frac{n-1}{2}}\frac{{\lambda }_i+{\lambda }_j+n-i-j}{n-i-j}}$

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

${\displaystyle \dim U_{\lambda }=\prod _{(i,j)\in \lambda ,i\ge j}\frac{n+{\lambda }_i+{\lambda }_j-i-j}{h_{\lambda }(i,j)}\prod _{(i,j)\in \lambda ,i<j}\frac{n-{\tilde{\lambda }}_i-{\tilde{\lambda }}_j+i+j-2}{h_{\lambda }(i,j)}}$

where ${\displaystyle {\lambda }_i,{\tilde{\lambda }}_i,h_{\lambda }(i,j)}$ 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,

${\displaystyle \dim U_{(k)}=\dim V_{(k)}-\dim V_{(k-2)}=(\genfrac{}{}{0ex}{}{n+k-1}{k})-(\genfrac{}{}{0ex}{}{n+k-3}{k})}$

Tensor products

In the stable range ${\displaystyle |\mu |+|\nu |\le \left[\frac{n}{2}\right]}$, 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 ${\displaystyle n\ge 12}$, we have

${\displaystyle \begin{array}{rl}[1]\otimes [1]& =[2]+[11]+[]\\ [1]\otimes [2]& =[21]+[3]+[1]\\ [1]\otimes [11]& =[111]+[21]+[1]\\ [1]\otimes [21]& =[31]+[22]+[211]+[2]+[11]\\ [1]\otimes [3]& =[4]+[31]+[2]\\ [2]\otimes [2]& =[4]+[31]+[22]+[2]+[11]+[]\\ [2]\otimes [11]& =[31]+[211]+[2]+[11]\\ [11]\otimes [11]& =[1111]+[211]+[22]+[2]+[11]+[]\\ [21]\otimes [3]& =[321]+[411]+[42]+[51]+[211]+[22]+2[31]+[4]+[11]+[2]\end{array}}$

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 ${\displaystyle 2|\nu |,{\tilde{\lambda }}_1+{\tilde{\lambda }}_2\le n}$. 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]

${\displaystyle \dim W_{\lambda }=\prod _{i=1}^n\frac{{\lambda }_i+n-i+1}{n-i+1}\prod _{1\le i<j\le n}\frac{{\lambda }_i-{\lambda }_j+j-i}{j-i}\cdot \frac{{\lambda }_i+{\lambda }_j+2n-i-j+2}{2n-i-j+2}}$

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

${\displaystyle \dim W_{\lambda }=\prod _{(i,j)\in \lambda ,i>j}\frac{n+{\lambda }_i+{\lambda }_j-i-j+2}{h_{\lambda }(i,j)}\prod _{(i,j)\in \lambda ,i\le j}\frac{n-{\tilde{\lambda }}_i-{\tilde{\lambda }}_j+i+j}{h_{\lambda }(i,j)}}$

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.

External links

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

References

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