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]

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

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

Tensor representation of ${\displaystyle GL(n,\mathbb {C} )}$	Dimension	Young diagram
Trivial representation	${\displaystyle 1}$	${\displaystyle ()}$
Determinant representation	${\displaystyle 1}$	${\displaystyle (1^{n})}$
Defining representation ${\displaystyle V}$	${\displaystyle n}$	${\displaystyle (1)}$
Symmetric representation ${\displaystyle {\text{Sym}}^{k}V}$	${\displaystyle {\binom {n+k-1}{k}}}$	${\displaystyle (k)}$
Antisymmetric representation ${\displaystyle \Lambda ^{k}V}$	${\displaystyle {\binom {n}{k}}}$	${\displaystyle (1^{k})}$

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 ${\displaystyle V^{\otimes r}\otimes (V^{*})^{\otimes s}}$, where ${\displaystyle V^{*}}$ 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 ${\displaystyle \lambda _{1}\geq \dots \geq \lambda _{n}}$. If ${\displaystyle \lambda _{k}\geq 0,\lambda _{k+1}\leq 0}$, 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,

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

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

• 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 ${\displaystyle \lambda _{n}+k\geq 0}$, 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 ${\displaystyle d_{\lambda }=\prod _{1\leq i<j\leq r}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}}$.^[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 ${\displaystyle V_{\lambda },V_{\lambda '}}$ 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 ${\displaystyle \forall i,\ \lambda _{i}-\lambda '_{i}=k}$.^[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 ${\displaystyle {\mathfrak {u}}(n)=\{a\in {\mathcal {M}}(n,\mathbb {C} ),a^{\dagger }+a=0\}}$ 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 ${\displaystyle U(n)\rightarrow GL(n,\mathbb {C} )}$. ^[5]

Tensor products

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

${\displaystyle V_{\lambda _{1}\mu _{1}}\otimes V_{\lambda _{2}\mu _{2}}=\bigoplus _{\nu ,\rho }V_{\nu \rho }^{\oplus \Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }},}$

where ${\displaystyle \Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }=0}$ unless ${\displaystyle |\nu |\leq |\lambda _{1}|+|\lambda _{2}|}$ and ${\displaystyle |\rho |\leq |\mu _{1}|+|\mu _{2}|}$. Calling ${\displaystyle l(\lambda )}$ the number of lines in a tableau, if ${\displaystyle l(\lambda _{1})+l(\lambda _{2})+l(\mu _{1})+l(\mu _{2})\leq n}$, then

${\displaystyle \Gamma _{\lambda _{1}\mu _{1},\lambda _{2}\mu _{2}}^{\nu \rho }=\sum _{\alpha ,\beta ,\eta ,\theta }\left(\sum _{\kappa }c_{\kappa ,\alpha }^{\lambda _{1}}c_{\kappa ,\beta }^{\mu _{2}}\right)\left(\sum _{\gamma }c_{\gamma ,\eta }^{\lambda _{2}}c_{\gamma ,\theta }^{\mu _{1}}\right)c_{\alpha ,\theta }^{\nu }c_{\beta ,\eta }^{\rho },}$

where the natural integers ${\displaystyle c_{\lambda ,\mu }^{\nu }}$ are Littlewood-Richardson coefficients.

Below are a few examples of such tensor products:

${\displaystyle R_{1}}$	${\displaystyle R_{2}}$	Tensor product ${\displaystyle R_{1}\otimes R_{2}}$
${\displaystyle V_{\lambda ()}}$	${\displaystyle V_{\mu ()}}$	${\displaystyle \sum _{\nu }c_{\lambda \mu }^{\nu }V_{\nu ()}}$
${\displaystyle V_{\lambda ()}}$	${\displaystyle V_{()\mu }}$	${\displaystyle \sum _{\kappa ,\nu ,\rho }c_{\kappa \nu }^{\lambda }c_{\kappa \rho }^{\mu }V_{\nu \rho }}$
${\displaystyle V_{()(1)}}$	${\displaystyle V_{(1)()}}$	${\displaystyle V_{(1)(1)}+V_{()()}}$
${\displaystyle V_{()(1)}}$	${\displaystyle V_{(k)()}}$	${\displaystyle V_{(k)(1)}+V_{(k-1)()}}$
${\displaystyle V_{(1)()}}$	${\displaystyle V_{(k)()}}$	${\displaystyle V_{(k+1)()}+V_{(k,1)()}}$
${\displaystyle V_{(1)(1)}}$	${\displaystyle V_{(1)(1)}}$	${\displaystyle V_{(2)(2)}+V_{(2)(11)}+V_{(11)(2)}+V_{(11)(11)}+2V_{(1)(1)}+V_{()()}}$

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 ${\displaystyle U_{(1^{n})}\otimes U_{\lambda }=U_{\lambda '}}$ where ${\displaystyle \lambda '}$ 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 ${\displaystyle {\tilde {\lambda }}_{1}\leq {\frac {n-1}{2}}}$ rows.
• For ${\displaystyle n}$ even, ${\displaystyle U_{\lambda }}$ is still irreducible as an ${\displaystyle SO(n,\mathbb {C} )}$ representation if ${\displaystyle {\tilde {\lambda }}_{1}\leq {\frac {n}{2}}-1}$, 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 ${\displaystyle (k\geq 0),(k\geq 1,1),(1,1,1)}$. The irreducible representations of ${\displaystyle SO(3,\mathbb {C} )}$ correspond to ${\displaystyle (k\geq 0)}$, 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]

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

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

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

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

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,

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

Tensor products

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

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

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 ${\displaystyle 2|\nu |,{\tilde {\lambda }}_{1}+{\tilde {\lambda }}_{2}\leq n}$. 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]

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]

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

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

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

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.