Representations of classical Lie groups

Group theory → 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

* List of simple Lie groups

Classical

* An
* Bn
* Cn
* Dn

Exceptional

* G2
* F4
* E6
* E7
* E8

Other Lie groups

* Circle
* Lorentz
* Poincaré
* Conformal group
* Diffeomorphism
* Loop
* Euclidean

Lie algebras

* Exponential map
* Adjoint representation

* (group
* algebra)

*     * Killing form
* Index

* Lie point symmetry
* Simple Lie algebra

Semisimple Lie algebra

* Dynkin diagrams
* Cartan subalgebra
*     * Root system
* Weyl group

*     * Real form
* Complexification

* Split Lie algebra
* Compact Lie algebra

Representation theory

* Lie group representation
* Lie algebra representation

Lie groups in physics

* Particle physics and representation theory
* Lorentz group representations
* Poincaré group representations
* Galilean group representations

Scientists

* Sophus Lie
* Henri Poincaré
* Wilhelm Killing
* Élie Cartan
* Hermann Weyl
* Claude Chevalley
* Harish-Chandra
* Armand Borel

* Glossary
* Table of Lie groups

* v
* t
* e

In mathematics, the finite-dimensional representations of the complex classical Lie groups GL(n,ℂ), SL(n,ℂ), O(n,ℂ), SO(n,ℂ), Sp(2n,ℂ), can be constructed using the general representation theory of semisimple Lie algebras. The groups SL(n,ℂ), SO(n,ℂ), Sp(2n,ℂ) are indeed simple Lie groups, and their finite-dimensional representations coincide[1] with those of their maximal compact subgroups, respectively SU(n), SO(n), Sp(n). In the classification of simple Lie algebras, the corresponding algebras are

SL(n,ℂ)->An−1SO(nodd,ℂ)->Bn−12SO(neven,ℂ)->Dn2Sp(2n,ℂ)->Cn

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.

## Contents

* 1 General linear group, special linear group and unitary group
* 1.1 Weyl's construction of tensor representations
* 1.2 General irreducible representations
* 1.3 Case of the special linear group
* 1.4 Case of the unitary group
* 1.5 Tensor products
* 2 Orthogonal group and special orthogonal group
* 2.1 Construction of representations
* 2.2 Case of the special orthogonal group
* 2.3 Dimensions
* 2.4 Tensor products
* 2.5 Branching rules from the general linear group
* 3 Symplectic group
* 3.1 Representations
* 3.2 Tensor products
* 4 External links
* 5 References

## General linear group, special linear group and unitary group

### Weyl's construction of tensor representations

Let V=ℂn be the defining representation of the general linear group GL(n,ℂ). Tensor representations are the subrepresentations of V⊗k (these are sometimes called polynomial representations). The irreducible subrepresentations of V⊗k are the images of V by Schur functors 𝕊λ associated to integer partitions λ of k into at most n integers, i.e. to Young diagrams of size λ1+⋯+λn=k with λn+1=0. (If λn+1>0 then 𝕊λ(V)=0.) Schur functors are defined using Young symmetrizers of the symmetric group Sk, which acts naturally on V⊗k. We write Vλ=𝕊λ(V).

The dimensions of these irreducible representations are[1]

dim⁡Vλ=∏1≤i<j≤nλi−λj+j−ij−i=∏(i,j)∈λn−i+jhλ(i,j)

where hλ(i,j) is the hook length of the cell (i,j) in the Young diagram λ.

* 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] χλ(g)=sλ(x1,…,xn) where x1,…,xn are the eigenvalues of g∈GL(n,ℂ).
* The second formula for the dimension is sometimes called Stanley's hook content formula.[2]

Examples of tensor representations:

Tensor representation of GL(n,ℂ) | Dimension  | Young diagram

|
|

Trivial representation  | 1 | ()
Determinant representation  | 1 | (1n)
Defining representation V | n | (1)
Symmetric representation SymkV | (n+k−1k) | (k)
Antisymmetric representation ΛkV | (nk) | (1k)

### General irreducible representations

Not all irreducible representations of GL(n,ℂ) are tensor representations. In general, irreducible representations of GL(n,ℂ) are mixed tensor representations, i.e. subrepresentations of V⊗r⊗(V*)⊗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,ℂ) is labeled by non increasing sequences of n integers λ1≥…≥λn. If λk≥0,λk+1≤0, we can associate to (λ1,…,λn) the pair of Young tableaux ([λ1…λk],[−λn,…,−λk+1]). This shows that irreducible representations of GL(n,ℂ) can be labeled by pairs of Young tableaux . Let us denote Vλμ=Vλ1,…,λn the irreducible representation of GL(n,ℂ) corresponding to the pair (λ,μ) or equivalently to the sequence (λ1,…,λn). With these notations,

* Vλ=Vλ(),V=V(1)()

* (Vλμ)*=Vμλ

* For k∈ℤ, denoting Dk the one-dimensional representation in which GL(n,ℂ) acts by (det⁡)k, Vλ1,…,λn=Vλ1+k,…,λn+k⊗D−k. If k is large enough that λn+k≥0, this gives an explicit description of Vλ1,…,λn in terms of a Schur functor.

* The dimension of Vλμ where λ=(λ1,…,λr),μ=(μ1,…,μs) is

dim⁡(Vλμ)=dλdμ∏i=1r(1−i−s+n)λi(1−i+r)λi∏j=1s(1−j−r+n)μi(1−j+s)μi∏i=1r∏j=1sn+1+λi+μj−i−jn+1−i−j where dλ=∏1≤i<j≤rλi−λj+j−ij−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 Vλ,Vλ′ of GL(n,ℂ) are equivalent as representations of the special linear group SL(n,ℂ) if and only if there is k∈ℤ such that ∀i, λi−λ'i=k.[1] For instance, the determinant representation V(1n) is trivial in SL(n,ℂ), i.e. it is equivalent to V(). In particular, irreducible representations of SL(n,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 GL(n,ℂ). The complexification of its Lie algebra 𝔲(n)={a∈ℳ(n,ℂ),a†+a=0} is the algebra 𝔤𝔩(n,ℂ). In Lie theoretic terms, U(n) is the compact real form of GL(n,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)->GL(n,ℂ). [5]

### Tensor products

Tensor products of finite-dimensional representations of GL(n,ℂ) are given by the following formula:[6]

Vλ1μ1⊗Vλ2μ2=⨁ν,ρVνρ⊕Γλ1μ1,λ2μ2νρ,

where Γλ1μ1,λ2μ2νρ=0 unless |ν|≤|λ1|+|λ2| and |ρ|≤|μ1|+|μ2|. Calling l(λ) the number of lines in a tableau, if l(λ1)+l(λ2)+l(μ1)+l(μ2)≤n, then

Γλ1μ1,λ2μ2νρ=∑α,β,η,θ(∑κcκ,αλ1cκ,βμ2)(∑γcγ,ηλ2cγ,θμ1)cα,θνcβ,ηρ,

where the natural integers cλ,μν are Littlewood-Richardson coefficients.

Below are a few examples of such tensor products:

R1 | R2 | Tensor product R1⊗R2

|
|

Vλ() | Vμ() | ∑νcλμνVν()
Vλ() | V()μ | ∑κ,ν,ρcκνλcκρμVνρ
V()(1) | V(1)() | V(1)(1)+V()()
V()(1) | V(k)() | V(k)(1)+V(k−1)()
V(1)() | V(k)() | V(k+1)()+V(k,1)()
V(1)(1) | V(1)(1) | V(2)(2)+V(2)(11)+V(11)(2)+V(11)(11)+2V(1)(1)+V()()

In the case of tensor representations, 3-j symbols and 6-j symbols are known.[7]

## Orthogonal group and special orthogonal group

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

### Construction of representations

Since O(n,ℂ) is a subgroup of GL(n,ℂ), any irreducible representation of GL(n,ℂ) is also a representation of O(n,ℂ), which may however not be irreducible. In order for a tensor representation of O(n,ℂ) to be irreducible, the tensors must be traceless.[8]

Irreducible representations of O(n,ℂ) are parametrized by a subset of the Young diagrams associated to irreducible representations of GL(n,ℂ): the diagrams such that the sum of the lengths of the first two columns is at most n.[8] The irreducible representation Uλ that corresponds to such a diagram is a subrepresentation of the corresponding GL(n,ℂ) representation Vλ. For example, in the case of symmetric tensors,[1]

V(k)=U(k)⊕V(k−2)

### Case of the special orthogonal group

The antisymmetric tensor U(1n) is a one-dimensional representation of O(n,ℂ), which is trivial for SO(n,ℂ). Then U(1n)⊗Uλ=Uλ′ where λ′ is obtained from λ by acting on the length of the first column as λ~1->n−λ~1.

* For n odd, the irreducible representations of SO(n,ℂ) are parametrized by Young diagrams with λ~1≤n−12 rows.
* For n even, Uλ is still irreducible as an SO(n,ℂ) representation if λ~1≤n2−1, but it reduces to a sum of two inequivalent SO(n,ℂ) representations if λ~1=n2.[8]

For example, the irreducible representations of O(3,ℂ) correspond to Young diagrams of the types (k≥0),(k≥1,1),(1,1,1). The irreducible representations of SO(3,ℂ) correspond to (k≥0), and dim⁡U(k)=2k+1. On the other hand, the dimensions of the spin representations of SO(3,ℂ) are even integers.[1]

### Dimensions

The dimensions of irreducible representations of SO(n,ℂ) are given by a formula that depends on the parity of n:[4]

(n even)dim⁡Uλ=∏1≤i<j≤n2λi−λj−i+j−i+j⋅λi+λj+n−i−jn−i−j
(n odd)dim⁡Uλ=∏1≤i<j≤n−12λi−λj−i+j−i+j∏1≤i≤j≤n−12λi+λj+n−i−jn−i−j

There is also an expression as a factorized polynomial in n:[4]

dim⁡Uλ=∏(i,j)∈λ, i≥jn+λi+λj−i−jhλ(i,j)∏(i,j)∈λ, i<jn−λ~i−λ~j+i+j−2hλ(i,j)

where λi,λ~i,hλ(i,j) are respectively row lengths, column lengths and hook lengths. In particular, antisymmetric representations have the same dimensions as their GL(n,ℂ) counterparts, dim⁡U(1k)=dim⁡V(1k), but symmetric representations do not,

dim⁡U(k)=dim⁡V(k)−dim⁡V(k−2)=(n+k−1k)−(n+k−3k)

### Tensor products

In the stable range |μ|+|ν|≤[n2], the tensor product multiplicities that appear in the tensor product decomposition Uλ⊗Uμ=⊕νNλ,μ,νUν are Newell-Littlewood numbers, which do not depend on n.[9] Beyond the stable range, the tensor product multiplicities become n-dependent modifications of the Newell-Littlewood numbers.[10][9][11] For example, for n≥12, we have

[1]⊗[1]=[2]+[11]+[][1]⊗[2]=[21]+[3]+[1][1]⊗[11]=[111]+[21]+[1][1]⊗[21]=[31]+[22]+[211]+[2]+[11][1]⊗[3]=[4]+[31]+[2][2]⊗[2]=[4]+[31]+[22]+[2]+[11]+[][2]⊗[11]=[31]+[211]+[2]+[11][11]⊗[11]=[1111]+[211]+[22]+[2]+[11]+[][21]⊗[3]=[321]+[411]+[42]+[51]+[211]+[22]+2[31]+[4]+[11]+[2]

### Branching rules from the general linear group

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 cλ,μν by the Littlewood restriction rule[12]

VνGL(n)=∑λ,μcλ,2μνUλO(n)

where 2μ is a partition into even integers. The rule is valid in the stable range 2|ν|,λ~1+λ~2≤n. The generalization to mixed tensor representations is

VλμGL(n)=∑α,β,γ,δcα,2γλcβ,2δμcα,βνUνO(n)

Similar branching rules can be written for the symplectic group.[12]

## Symplectic group

### Representations

The finite-dimensional irreducible representations of the symplectic group Sp(2n,ℂ) are parametrized by Young diagrams with at most n rows. The dimension of the corresponding representation is[8]

dim⁡Wλ=∏i=1nλi+n−i+1n−i+1∏1≤i<j≤nλi−λj+j−ij−i⋅λi+λj+2n−i−j+22n−i−j+2

There is also an expression as a factorized polynomial in n:[4]

dim⁡Wλ=∏(i,j)∈λ, i>jn+λi+λj−i−j+2hλ(i,j)∏(i,j)∈λ, i≤jn−λ~i−λ~j+i+jhλ(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

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 , Wikidata Q55865630
2. ↑ Hawkes, Graham (2013-10-19). "An Elementary Proof of the Hook Content Formula". arXiv:1310.5919v2 [math.CO].
3. ↑ 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). doi:10.1007/JHEP04(2020)117. Bibcode: 2020JHEP...04..117B.
4. ↑ 4.0 4.1 4.2 4.3 , Wikidata Q104601301
5. ↑ Cvitanović, Predrag (2008). Group theory: Birdtracks, Lie's, and exceptional groups. https://birdtracks.eu/.
6. ↑ 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.
7. ↑ Artamonov, Dmitry (2025). "Calculation of $ 6j $-Symbols for the Lie Algebra $ {\mathfrak{gl}}_{n} $". Siberian Mathematical Journal 66 (4): 875–890. doi:10.1134/S0037446625040019.
8. ↑ 8.0 8.1 8.2 8.3 Hamermesh, Morton (1989). Group theory and its application to physical problems. New York: Dover Publications. ISBN 0-486-66181-4. OCLC 20218471.
9. ↑ 9.0 9.1 Gao, Shiliang; Orelowitz, Gidon; Yong, Alexander (2021). "Newell-Littlewood numbers". Transactions of the American Mathematical Society 374 (9): 6331–6366. doi:10.1090/tran/8375.
10. ↑ Steven Sam (2010-01-18). "Littlewood-Richardson coefficients for classical groups". https://concretenonsense.wordpress.com/2010/01/18/littlewood-richardson-coefficients-for-classical-groups/.
11. ↑ , Wikidata Q56443390
12. ↑ 12.0 12.1 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. doi:10.1090/S0002-9947-04-03722-5.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Representations of classical Lie groups. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Representations_of_classical_Lie_groups&oldid=4152449"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template