Representations of classical Lie groups#Weyl's construction of tensor representations

Let $V=\backslash mathbb\{C\}^n$ be the defining representation of the general linear group $GL(n,\backslash mathbb\{C\})$. Tensor representations are the subrepresentations of $V^\{\backslash otimes\; k\}$ (these are sometimes called polynomial representations). The irreducible subrepresentations of $V^\{\backslash otimes\; k\}$ are the images of $V$ by Schur functors $\backslash mathbb\{S\}^\backslash lambda$ associated to integer partitions $\backslash lambda$ of $k$ into at most $n$ integers, i.e. to Young diagrams of size $\backslash lambda\_1+\backslash cdots\; +\; \backslash lambda\_n\; =\; k$ with $\backslash lambda\_\{n+1\}=0$. (If $\backslash lambda\_\{n+1\}>0$ then $\backslash mathbb\{S\}^\backslash lambda(V)=0$.) Schur functors are defined using Young symmetrizers of the symmetric group $S\_k$, which acts naturally on $V^\{\backslash otimes\; k\}$. We write $V\_\backslash lambda\; =\; \backslash mathbb\{S\}^\backslash lambda(V)$.

The dimensions of these irreducible representations are{{cite Q|Q55865630}}

:$$

\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 $h\_\backslash lambda(i,j)$ is the hook length of the cell $(i,j)$ in the Young diagram $\backslash lambda$.

• The first formula for the dimension is a special case of a formula that gives the characters of representations in terms of Schur polynomials,{{cite Q|Q55865630}} $$

\chi_\lambda(g) = s_\lambda(x_1,\dots, x_n)

where $x\_1,\backslash dots\; ,x\_n$ are the eigenvalues of $g\backslash in\; GL(n,\backslash mathbb\{C\})$.

• The second formula for the dimension is sometimes called Stanley's hook content formula.

Examples of tensor representations:

Not all irreducible representations of $GL(n,\backslash mathbb\; C)$ are tensor representations. In general, irreducible representations of $GL(n,\backslash mathbb\; C)$ are mixed tensor representations, i.e. subrepresentations of $V^\{\backslash otimes\; r\}\; \backslash otimes\; (V^*)^\{\backslash otimes\; s\}$, where $V^*$ is the dual representation of $V$ (these are sometimes called rational representations). In the end, the set of irreducible representations of $GL(n,\backslash mathbb\; C)$ is labeled by non increasing sequences of $n$ integers $\backslash lambda\_1\backslash geq\; \backslash dots\; \backslash geq\; \backslash lambda\_n$.

If $\backslash lambda\_k\; \backslash geq\; 0,\; \backslash lambda\_\{k+1\}\; \backslash leq\; 0$, we can associate to $(\backslash lambda\_1,\; \backslash dots\; ,\backslash lambda\_n)$ the pair of Young tableaux $([\backslash lambda\_1\backslash dots\backslash lambda\_k],[-\backslash lambda\_n,\backslash dots,-\backslash lambda\_\{k+1\}])$. This shows that irreducible representations of $GL(n,\backslash mathbb\; C)$ can be labeled by pairs of Young tableaux . Let us denote $V\_\{\backslash lambda\backslash mu\}\; =\; V\_\{\backslash lambda\_1,\backslash dots,\backslash lambda\_n\}$ the irreducible representation of $GL(n,\backslash mathbb\; C)$ corresponding to the pair $(\backslash lambda,\backslash mu)$ or equivalently to the sequence $(\backslash lambda\_1,\backslash dots,\backslash lambda\_n)$. With these notations,

• $V\_\{\backslash lambda\}=V\_\{\backslash lambda()\},\; V\; =\; V\_\{(1)()\}$

• $(V\_\{\backslash lambda\backslash mu\})^*\; =\; V\_\{\backslash mu\backslash lambda\}$

• For $k\; \backslash in\; \backslash mathbb\; Z$, denoting $D\_k$ the one-dimensional representation in which $GL(n,\backslash mathbb\; C)$ acts by $(\backslash det)^k$, $V\_\{\backslash lambda\_1,\backslash dots,\backslash lambda\_n\}\; =\; V\_\{\backslash lambda\_1+k,\backslash dots,\backslash lambda\_n+k\}\; \backslash otimes\; D\_\{-k\}$. If $k$ is large enough that $\backslash lambda\_n\; +\; k\; \backslash geq\; 0$, this gives an explicit description of $V\_\{\backslash lambda\_1,\; \backslash dots,\backslash lambda\_n\}$ in terms of a Schur functor.

• The dimension of $V\_\{\backslash lambda\backslash mu\}$ where $\backslash lambda\; =\; (\backslash lambda\_1,\backslash dots,\backslash lambda\_r),\; \backslash mu=(\backslash mu\_1,\backslash dots,\backslash mu\_s)$ is

:$\backslash dim(V\_\{\backslash lambda\backslash mu\})\; =\; d\_\backslash lambda\; d\_\backslash mu\; \backslash prod\_\{i=1\}^r\; \backslash frac\{(1-i-s+n)\_\{\backslash lambda\_i\}\}\{(1-i+r)\_\{\backslash lambda\_i\}\}\; \backslash prod\_\{j=1\}^s\; \backslash frac\{(1-j-r+n)\_\{\backslash mu\_i\}\}\{(1-j+s)\_\{\backslash mu\_i\}\}\backslash prod\_\{i=1\}^r\; \backslash prod\_\{j=1\}^s\; \backslash frac\{n+1\; +\; \backslash lambda\_i\; +\; \backslash mu\_j\; -\; i-\; j\; \}\{n+1\; -i\; -j\; \}$

where .{{cite journal | last1=Binder, D. - Rychkov, S. |title=Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of O(N) Symmetry with Non-integer N |journal=Journal of High Energy Physics |year=2020 |volume=2020 |issue=4 |page=117 |doi=10.1007/JHEP04(2020)117|arxiv=1911.07895 |bibcode=2020JHEP...04..117B |doi-access=free }} See for an interpretation as a product of n-dependent factors divided by products of hook lengths.

Two representations $V\_\{\backslash lambda\},V\_\{\backslash lambda\text{'}\}$ of $GL(n,\backslash mathbb\{C\})$ are equivalent as representations of the special linear group $SL(n,\backslash mathbb\{C\})$ if and only if there is $k\backslash in\backslash mathbb\{Z\}$ such that $\backslash forall\; i,\backslash \; \backslash lambda\_i-\backslash lambda\text{'}\_i=k$.{{cite Q|Q55865630}} For instance, the determinant representation $V\_\{(1^n)\}$ is trivial in $SL(n,\backslash mathbb\{C\})$, i.e. it is equivalent to $V\_\{()\}$.

In particular, irreducible representations of $SL(n,\backslash mathbb\; C)$ can be indexed by Young tableaux, and are all tensor representations (not mixed).

The unitary group is the maximal compact subgroup of $GL(n,\backslash mathbb\; C)$. The complexification of its Lie algebra $\backslash mathfrak\; u(n)\; =\; \backslash \{a\; \backslash in\; \backslash mathcal\; M(n,\backslash mathbb\; C),\; a^\backslash dagger\; +\; a\; =\; 0\backslash \}$ is the algebra $\backslash mathfrak\{gl\}(n,\backslash mathbb\; C)$. In Lie theoretic terms, $U(n)$ is the compact real form of $GL(n,\backslash mathbb\; C)$, 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 $U(n)\; \backslash rightarrow\; GL(n,\backslash mathbb\; C)$. {{cite book |last1=Cvitanović |first1=Predrag |title=Group theory: Birdtracks, Lie's, and exceptional groups |date=2008 |url=https://birdtracks.eu/}}

Tensor products of finite-dimensional representations of $GL(n,\backslash mathbb\{C\})$ are given by the following formula:{{cite journal |last1=Koike |first1=Kazuhiko |title=On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters |journal=Advances in Mathematics |year=1989 |volume=74 |pages=57–86 |doi=10.1016/0001-8708(89)90004-2|doi-access=free }}

:$$

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}},

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

:$$

\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},

where the natural integers $c\_\{\backslash lambda,\backslash mu\}^\backslash nu$ 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.

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

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

Irreducible representations of $O(n,\backslash mathbb\{C\})$ are parametrized by a subset of the Young diagrams associated to irreducible representations of $GL(n,\backslash mathbb\{C\})$: the diagrams such that the sum of the lengths of the first two columns is at most $n$. The irreducible representation $U\_\backslash lambda$ that corresponds to such a diagram is a subrepresentation of the corresponding $GL(n,\backslash mathbb\{C\})$ representation $V\_\backslash lambda$. For example, in the case of symmetric tensors,{{cite Q|Q55865630}}

:$$

V_{(k)} = U_{(k)} \oplus V_{(k-2)}

The antisymmetric tensor $U\_\{(1^n)\}$ is a one-dimensional representation of $O(n,\backslash mathbb\{C\})$, which is trivial for $SO(n,\backslash mathbb\{C\})$. Then $U\_\{(1^n)\}\backslash otimes\; U\_\backslash lambda\; =\; U\_\{\backslash lambda\text{'}\}$ where $\backslash lambda\text{'}$ is obtained from $\backslash lambda$ by acting on the length of the first column as $\backslash tilde\{\backslash lambda\}\_1\backslash to\; n-\backslash tilde\{\backslash lambda\}\_1$.

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

For example, the irreducible representations of $O(3,\backslash mathbb\{C\})$ correspond to Young diagrams of the types $(k\backslash geq\; 0),(k\backslash geq\; 1,1),(1,1,1)$. The irreducible representations of $SO(3,\backslash mathbb\{C\})$ correspond to $(k\backslash geq\; 0)$, and $\backslash dim\; U\_\{(k)\}=2k+1$.

On the other hand, the dimensions of the spin representations of $SO(3,\backslash mathbb\{C\})$ are even integers.{{cite Q|Q55865630}}

The dimensions of irreducible representations of $SO(n,\backslash mathbb\{C\})$ are given by a formula that depends on the parity of $n$:{{cite Q|Q104601301}}

:$$

(n\text{ even}) \qquad \dim U_\lambda = \prod_{1\leq i

:$$

(n\text{ odd}) \qquad \dim U_\lambda = \prod_{1\leq i

\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 $n$:{{cite Q|Q104601301}}

:$$

\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 $\backslash lambda\_i,\backslash tilde\{\backslash lambda\}\_i,h\_\backslash lambda(i,j)$ are respectively row lengths, column lengths and hook lengths. In particular, antisymmetric representations have the same dimensions as their $GL(n,\backslash mathbb\{C\})$ counterparts, $\backslash dim\; U\_\{(1^k)\}=\backslash dim\; V\_\{(1^k)\}$, 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}

In the stable range $|\backslash mu|+|\backslash nu|\backslash leq\; \backslash left[\backslash frac\{n\}\{2\}\backslash right]$, the tensor product multiplicities that appear in the tensor product decomposition $U\_\backslash lambda\backslash otimes\; U\_\backslash mu\; =\; \backslash oplus\_\backslash nu\; N\_\{\backslash lambda,\backslash mu,\backslash nu\}\; U\_\backslash nu$ 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.{{CiteQ|Q56443390}} For example, for $n\backslash geq\; 12$, we have

:$$

\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}

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 $c\_\{\backslash lambda,\backslash mu\}^\backslash nu$ by the Littlewood restriction rule

:$$

V_\nu^{GL(n)} = \sum_{\lambda,\mu} c_{\lambda,2\mu}^\nu U_\lambda^{O(n)}

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

:$$

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.

The finite-dimensional irreducible representations of the symplectic group $Sp(2n,\backslash mathbb\{C\})$ are parametrized by Young diagrams with at most $n$ rows. The dimension of the corresponding representation is

:$$

\dim W_\lambda = \prod_{i=1}^n \frac{\lambda_i+n-i+1}{n-i+1} \prod_{1\leq i

There is also an expression as a factorized polynomial in $n$:{{cite Q|Q104601301}}

:$$

\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)}

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.

• [http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/form.html Lie online service], an online interface to the [http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/ Lie software].

{{Reflist|refs=

{{cite arXiv | last=Artamonov | first=Dmitry | title=Calculation of $6j$-symbols for the Lie algebra $\backslash mathfrak\{gl\}\_n$ | date=2024-05-09 | eprint=2405.05628 | class=math.RT}}

{{cite journal|last1=Howe|first1=Roger| authorlink=Roger Evans Howe|last2=Tan|first2=Eng-Chye|last3= Willenbring|first3= Jeb F.|authorlink3=Jeb Willenbring|title=Stable branching rules for classical symmetric pairs|journal=Transactions of the American Mathematical Society|volume= 357|issue=4|year=2005|pages= 1601–1626|doi=10.1090/S0002-9947-04-03722-5|doi-access=free|arxiv=math/0311159}}

{{cite arXiv | last=Hawkes | first=Graham | title=An Elementary Proof of the Hook Content Formula | date=2013-10-19 | class=math.CO |eprint=1310.5919v2}}

{{cite web | author = Steven Sam|title=Littlewood-Richardson coefficients for classical groups | website=Concrete Nonsense | date=2010-01-18 | url=https://concretenonsense.wordpress.com/2010/01/18/littlewood-richardson-coefficients-for-classical-groups/ | archive-url=https://web.archive.org/web/20190618084957/https://concretenonsense.wordpress.com/2010/01/18/littlewood-richardson-coefficients-for-classical-groups/ | archive-date=2019-06-18 | url-status=live | access-date=2021-01-05}}

{{cite journal | last1=Gao | first1=Shiliang | last2=Orelowitz | first2=Gidon | last3=Yong | first3=Alexander | title=Newell-Littlewood numbers | journal=Transactions of the American Mathematical Society | year=2021 | volume=374 | issue=9 | pages=6331–6366 | doi=10.1090/tran/8375 | arxiv=2005.09012v1| s2cid=218684561 }}

{{cite book | last=Hamermesh | first=Morton | title=Group theory and its application to physical problems | publisher=Dover Publications | publication-place=New York | year=1989 | isbn=0-486-66181-4 | oclc=20218471}}

}}

Category:Representation theory of Lie groups

Category:Lie groups