Representations of classical Lie groups

Lie groups
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
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Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
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Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
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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

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

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 and special linear group[]

Weyl's construction[]

Let ${\displaystyle V=\mathbb {C} ^{n}}$ be the defining representation of the general linear group ${\displaystyle GL(n,\mathbb {C} )}$. Any irreducible finite-dimensional representation of ${\displaystyle GL(n,\mathbb {C} )}$ is a tensor representation, i.e. a subrepresentation of ${\displaystyle V^{\otimes k}}$ for some integer ${\displaystyle k\in \mathbb {N} }$.

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 the 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[]

Irreducible representation of ${\displaystyle GL(n,\mathbb {C} )}$	Dimension	Young diagram
Trivial representation	${\displaystyle 1}$	${\displaystyle ()}$
Determinant representation	${\displaystyle 1}$	${\displaystyle (1^{n})}$
Defining representation ${\displaystyle V}$	${\displaystyle n}$	${\displaystyle (1)}$
Adjoint representation	${\displaystyle n^{2}-1}$	${\displaystyle (2,1^{n-2})}$
Symmetric representation ${\displaystyle {\text{Sym}}^{k}V}$	${\displaystyle {\binom {n+k-1}{k}}}$	${\displaystyle (k)}$
Antisymmetric representation ${\displaystyle \Lambda ^{k}V}$	${\displaystyle {\binom {n}{k}}}$	${\displaystyle (1^{k})}$

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] In particular, the determinant representation ${\displaystyle V_{(1^{n})}}$ is trivial in ${\displaystyle SL(n,\mathbb {C} )}$, i.e. it is equivalent to ${\displaystyle V_{()}}$.

Tensor products[]

Tensor products of finite-dimensional representations of ${\displaystyle GL(n,\mathbb {C} )}$ are^[1]

${\displaystyle V_{\lambda }\otimes V_{\mu }=\sum _{\nu }c_{\lambda \mu }^{\nu }V_{\nu }}$

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

${\displaystyle V_{(1)}\otimes V_{(k)}=V_{(k+1)}\oplus V_{(k,1)}}$

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.^[3]

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 a most ${\displaystyle n}$.^[3] 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 ${\displaystyle {\tilde {\lambda }}_{1}={\frac {n}{2}}}$.^[3]

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,

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

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 ${\displaystyle n}$.^[5] Beyond the stable range, the tensor product multiplicities become ${\displaystyle n}$-dependent modifications of the Newell-Littlewood numbers.^[6][5][7]

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^[3]

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

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.

External links[]

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

References[]