Lie groups and Lie algebras |
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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)
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- Circle
- Lorentz
- Poincaré
- Conformal group
- Diffeomorphism
- Loop
- Euclidean
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In mathematics, the finite-dimensional representations of the complex classical Lie groups
,
,
,
,
, can be constructed using the general representation theory of semisimple Lie algebras. The groups
,
,
are indeed simple Lie groups, and their finite-dimensional representations coincide[1] with those of their maximal compact subgroups, respectively
,
,
. 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}},\mathbb {C} )&\to D_{\frac {n}{2}}\\Sp(2n,\mathbb {C} )&\to C_{n}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bed93ae5842cbe8e06de0df6da819691fdf25c24)
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.
Weyl's construction of tensor representations
Let
be the defining representation of the general linear group
. Tensor representations are the subrepresentations of
(these are sometimes called polynomial representations). The irreducible subrepresentations of
are the images of
by Schur functors
associated to integer partitions
of
into at most
integers, i.e. to Young diagrams of size
with
. (If
then
.) Schur functors are defined using Young symmetrizers of the symmetric group
, which acts naturally on
. We write
.
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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c7ca1878d4441d3813f3829759b5c6fb46acfc)
where
is the hook length of the cell
in the Young diagram
.
- 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]
where
are the eigenvalues of
. - The second formula for the dimension is sometimes called Stanley's hook content formula.[2]
Examples of tensor representations:
Tensor representation of | Dimension | Young diagram |
Trivial representation | | |
Determinant representation | | |
Defining representation | | |
Symmetric representation | | |
Antisymmetric representation | | |
General irreducible representations
Not all irreducible representations of
are tensor representations. In general, irreducible representations of
are mixed tensor representations, i.e. subrepresentations of
, where
is the dual representation of
(these are sometimes called rational representations). In the end, the set of irreducible representations of
is labeled by non increasing sequences of
integers
. If
, we can associate to
the pair of Young tableaux
. This shows that irreducible representations of
can be labeled by pairs of Young tableaux . Let us denote
the irreducible representation of
corresponding to the pair
or equivalently to the sequence
. With these notations,
![{\displaystyle V_{\lambda }=V_{\lambda ()},V=V_{(1)()}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f992fd7e4dbc5a634ecd436964a423e35017286)
![{\displaystyle (V_{\lambda \mu })^{*}=V_{\mu \lambda }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1f43461bda134d0c84dd05258b83fe9544f622)
- For
, denoting
the one-dimensional representation in which
acts by
,
. If
is large enough that
, this gives an explicit description of
in terms of a Schur functor.
- The dimension of
where
is
where
.[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
of
are equivalent as representations of the special linear group
if and only if there is
such that
.[1] For instance, the determinant representation
is trivial in
, i.e. it is equivalent to
. In particular, irreducible representations of
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
. The complexification of its Lie algebra
is the algebra
. In Lie theoretic terms,
is the compact real form of
, 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
. [5]
Tensor products
Tensor products of finite-dimensional representations of
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 }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60b8d3ca727496d578f657de0638bfb36378ad0a)
where
unless
and
. Calling
the number of lines in a tableau, if
, 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 },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c714cb375bbe1de1fbd14fb5148c43bdd2516f2c)
where the natural integers
are Littlewood-Richardson coefficients.
Below are a few examples of such tensor products:
| | Tensor product |
| | |
| | |
| | |
| | |
| | |
| | |
In addition to the Lie group representations described here, the orthogonal group
and special orthogonal group
have spin representations, which are projective representations of these groups, i.e. representations of their universal covering groups.
Construction of representations
Since
is a subgroup of
, any irreducible representation of
is also a representation of
, which may however not be irreducible. In order for a tensor representation of
to be irreducible, the tensors must be traceless.[7]
Irreducible representations of
are parametrized by a subset of the Young diagrams associated to irreducible representations of
: the diagrams such that the sum of the lengths of the first two columns is at most
.[7] The irreducible representation
that corresponds to such a diagram is a subrepresentation of the corresponding
representation
. For example, in the case of symmetric tensors,[1]
![{\displaystyle V_{(k)}=U_{(k)}\oplus V_{(k-2)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf6a6e29fc8e82438863a27c6da1c0faf2009050)
Case of the special orthogonal group
The antisymmetric tensor
is a one-dimensional representation of
, which is trivial for
. Then
where
is obtained from
by acting on the length of the first column as
.
- For
odd, the irreducible representations of
are parametrized by Young diagrams with
rows. - For
even,
is still irreducible as an
representation if
, but it reduces to a sum of two inequivalent
representations if
.[7]
For example, the irreducible representations of
correspond to Young diagrams of the types
. The irreducible representations of
correspond to
, and
. On the other hand, the dimensions of the spin representations of
are even integers.[1]
Dimensions
The dimensions of irreducible representations of
are given by a formula that depends on the parity of
:[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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76e398fbc26e58732fb256391b44a3476600513a)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9eed2198f4afeed1c9a0dd2f2379280a36fd9a)
There is also an expression as a factorized polynomial in
:[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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e66c361fe52bcc1ddec606324ed901b689dadbeb)
where
are respectively row lengths, column lengths and hook lengths. In particular, antisymmetric representations have the same dimensions as their
counterparts,
, 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c9de670b114d45df1300cf3663794de39539f84)
Tensor products
In the stable range
, the tensor product multiplicities that appear in the tensor product decomposition
are Newell-Littlewood numbers, which do not depend on
.[8] Beyond the stable range, the tensor product multiplicities become
-dependent modifications of the Newell-Littlewood numbers.[9][8][10] For example, for
, 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/611f839bd19d2cce19a1677af0645f05c9e283f2)
Branching rules from the general linear group
Since the orthogonal group is a subgroup of the general linear group, representations of
can be decomposed into representations of
. The decomposition of a tensor representation is given in terms of Littlewood-Richardson coefficients
by the Littlewood restriction rule[11]
![{\displaystyle V_{\nu }^{GL(n)}=\sum _{\lambda ,\mu }c_{\lambda ,2\mu }^{\nu }U_{\lambda }^{O(n)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/727266b60cf898f19b8c33f9b56f4f95020325ee)
where
is a partition into even integers. The rule is valid in the stable range
. 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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942ea1d08c9f18473f070acaaff246a6ab15d953)
Similar branching rules can be written for the symplectic group.[11]
Representations
The finite-dimensional irreducible representations of the symplectic group
are parametrized by Young diagrams with at most
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/429d9cf604b192d064ad3620c33307a90a4cefcd)
There is also an expression as a factorized polynomial in
:[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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2989824d4739cc3a4f3db653d22552be6143d4)
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
- ^ a b c d e f William Fulton; Joe Harris (2004). "Representation Theory". Graduate Texts in Mathematics. doi:10.1007/978-1-4612-0979-9. ISSN 0072-5285. Wikidata Q55865630.
- ^ Hawkes, Graham (2013-10-19). "An Elementary Proof of the Hook Content Formula". arXiv:1310.5919v2 [math.CO].
- ^ Binder, D. - Rychkov, S. (2020). "Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of O(N) Symmetry with Non-integer N". Journal of High Energy Physics. 2020 (4): 117. arXiv:1911.07895. Bibcode:2020JHEP...04..117B. doi:10.1007/JHEP04(2020)117.
{{cite journal}}
: CS1 maint: multiple names: authors list (link) - ^ a b c d N El Samra; R C King (December 1979). "Dimensions of irreducible representations of the classical Lie groups". Journal of Physics A. 12 (12): 2317–2328. doi:10.1088/0305-4470/12/12/010. ISSN 1751-8113. Zbl 0445.22020. Wikidata Q104601301.
- ^ Cvitanović, Predrag (2008). Group theory: Birdtracks, Lie's, and exceptional groups.
- ^ Koike, Kazuhiko (1989). "On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters". Advances in Mathematics. 74: 57–86. doi:10.1016/0001-8708(89)90004-2.
- ^ a b c d Hamermesh, Morton (1989). Group theory and its application to physical problems. New York: Dover Publications. ISBN 0-486-66181-4. OCLC 20218471.
- ^ a b Gao, Shiliang; Orelowitz, Gidon; Yong, Alexander (2021). "Newell-Littlewood numbers". Transactions of the American Mathematical Society. 374 (9): 6331–6366. arXiv:2005.09012v1. doi:10.1090/tran/8375. S2CID 218684561.
- ^ Steven Sam (2010-01-18). "Littlewood-Richardson coefficients for classical groups". Concrete Nonsense. Archived from the original on 2019-06-18. Retrieved 2021-01-05.
- ^ Kazuhiko Koike; Itaru Terada (May 1987). "Young-diagrammatic methods for the representation theory of the classical groups of type Bn, Cn, Dn". Journal of Algebra. 107 (2): 466–511. doi:10.1016/0021-8693(87)90099-8. ISSN 0021-8693. Zbl 0622.20033. Wikidata Q56443390.
- ^ a b Howe, Roger; Tan, Eng-Chye; Willenbring, Jeb F. (2005). "Stable branching rules for classical symmetric pairs". Transactions of the American Mathematical Society. 357 (4): 1601–1626. arXiv:math/0311159. doi:10.1090/S0002-9947-04-03722-5.