Tensor product of representations

In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few.

Definition

Group representations

If ${\displaystyle V_{1},V_{2}}$ are linear representations of a group ${\displaystyle G}$, then their tensor product is the tensor product of vector spaces ${\displaystyle V_{1}\otimes V_{2}}$ with the linear action of ${\displaystyle G}$ uniquely determined by the condition that

${\displaystyle g\cdot (v_{1}\otimes v_{2})=(g\cdot v_{1})\otimes (g\cdot v_{2})}$[1][2]

for all ${\displaystyle v_{1}\in V_{1}}$ and ${\displaystyle v_{2}\in V_{2}}$. Although not every element of ${\displaystyle V_{1}\otimes V_{2}}$ is expressible in the form ${\displaystyle v_{1}\otimes v_{2}}$, the universal property of the tensor product operation guarantees that this action is well defined.

In the language of homomorphisms, if the actions of ${\displaystyle G}$ on ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$ are given by homomorphisms ${\displaystyle \Pi _{1}:G\rightarrow \operatorname {GL} (V_{1})}$ and ${\displaystyle \Pi _{2}:G\rightarrow \operatorname {GL} (V_{2})}$, then the tensor product representation is given by the homomorphism ${\displaystyle \Pi _{1}\otimes \Pi _{2}:G\rightarrow \operatorname {GL} (V_{1}\otimes V_{2})}$ given by

${\displaystyle \Pi _{1}\otimes \Pi _{2}(g)=\Pi _{1}(g)\otimes \Pi _{2}(g)}$,

where ${\displaystyle \Pi _{1}(g)\otimes \Pi _{2}(g)}$ is the tensor product of linear maps.[3]

One can extend the notion of tensor products to any finite number of representations. If V is a linear representation of a group G, then with the above linear action, the tensor algebra ${\displaystyle T(V)}$ is an algebraic representation of G; i.e., each element of G acts as an algebra automorphism.

Lie algebra representations

If ${\displaystyle (V_{1},\pi _{1})}$ and ${\displaystyle (V_{2},\pi _{2})}$ are representations of a Lie algebra ${\displaystyle {\mathfrak {g}}}$, then the tensor product of these representations is given by the map ${\displaystyle \pi _{1}\otimes \pi _{2}:{\mathfrak {g}}\rightarrow \operatorname {End} (V_{1}\otimes V_{2})}$ given by[4]

${\displaystyle \pi _{1}\otimes \pi _{2}(X)=\pi _{1}(X)\otimes I+I\otimes \pi _{2}(X)}$.

The motivation for this definition comes from the case in which ${\displaystyle \pi _{1}}$ and ${\displaystyle \pi _{2}}$ come from representations ${\displaystyle \Pi _{1}}$ and ${\displaystyle \Pi _{2}}$ of a Lie group ${\displaystyle G}$. In that case, a simple computation shows that the Lie algebra representation associated to ${\displaystyle \Pi _{1}\otimes \Pi _{2}}$ is given by the preceding formula.[5]

Action on linear maps

If ${\displaystyle (V_{1},\Pi _{1})}$ and ${\displaystyle (V_{2},\Pi _{2})}$ are representations of a group ${\displaystyle G}$, let ${\displaystyle \operatorname {Hom} (V_{1},V_{2})}$ denote the space of all linear maps from ${\displaystyle V_{1}}$ to ${\displaystyle V_{2}}$. Then ${\displaystyle \operatorname {Hom} (V_{1},V_{2})}$ can be given the structure of a representation by defining

${\displaystyle g\cdot A=\Pi _{2}(g)A\Pi _{1}(g)^{-1}}$

for all ${\displaystyle A\in \operatorname {Hom} (V,W)}$. Now, there is a natural isomorphism

${\displaystyle \operatorname {Hom} (V,W)\cong V^{*}\otimes W}$

as vector spaces;[2] this vector space isomorphism is in fact an isomorphism of representations.[6]

The trivial subrepresentation ${\displaystyle \operatorname {Hom} (V,W)^{G}}$ consists of G-linear maps; i.e.,

${\displaystyle \operatorname {Hom} _{G}(V,W)=\operatorname {Hom} (V,W)^{G}.}$

Let ${\displaystyle E=\operatorname {End} (V)}$ denote the endomorphism algebra of V and let A denote the subalgebra of ${\displaystyle E^{\otimes m}}$ consisting of symmetric tensors. The main theorem of invariant theory states that A is semisimple when the characteristic of the base field is zero.

Clebsch–Gordan theory

The general problem

The tensor product of two irreducible representations ${\displaystyle V_{1},V_{2}}$ of a group or Lie algebra is usually not irreducible. It is therefore of interest to attempt to decompose ${\displaystyle V_{1}\otimes V_{2}}$ into irreducible pieces. This decomposition problem is known as the Clebsch–Gordan problem.

The SU(2) case

The prototype example of this problem is the case of the rotation group SO(3)—or its double cover, the special unitary group SU(2). The irreducible representations of SU(2) are described by a parameter ${\displaystyle \ell }$, whose possible values are

${\displaystyle \ell =0,1/2,1,3/2,\ldots .}$

(The dimension of the representation is then ${\displaystyle 2\ell +1}$.) Let us take two parameters ${\displaystyle \ell }$ and ${\displaystyle m}$ with ${\displaystyle \ell \geq m}$. Then the tensor product representation ${\displaystyle V_{\ell }\otimes V_{m}}$ then decomposes as follows:[7]

${\displaystyle V_{\ell }\otimes V_{m}\cong V_{\ell +m}\oplus V_{\ell +m-1}\oplus \cdots \oplus V_{\ell -m+1}\oplus V_{\ell -m}.}$

Consider, as an example, the tensor product of the four-dimensional representation ${\displaystyle V_{3/2}}$ and the three-dimensional representation ${\displaystyle V_{1}}$. The tensor product representation ${\displaystyle V_{3/2}\otimes V_{1}}$ has dimension 12 and decomposes as

${\displaystyle V_{3/2}\otimes V_{1}\cong V_{5/2}\oplus V_{3/2}\oplus V_{1/2}}$,

where the representations on the right-hand side have dimension 6, 4, and 2, respectively. We may summarize this result arithmetically as ${\displaystyle 4\times 3=6+4+2}$.

The SU(3) case

In the case of the group SU(3), all the irreducible representations can be generated from the standard 3-dimensional representation and its dual, as follows. To generate the representation with label ${\displaystyle (m_{1},m_{2})}$, one takes the tensor product of ${\displaystyle m_{1}}$ copies of the standard representation and ${\displaystyle m_{2}}$ copies of the dual of the standard representation, and then takes the invariant subspace generated by the tensor product of the highest weight vectors.[8]

In contrast to the situation for SU(2), in the Clebsch–Gordan decomposition for SU(3), a given irreducible representation ${\displaystyle W}$ may occur more than once in the decomposition of ${\displaystyle U\otimes V}$.

Tensor power

As with vector spaces, one can define the kth tensor power of a representation V to be the vector space ${\displaystyle V^{\otimes k}}$ with the action given above.

The symmetric and alternating square

Over a field of characteristic zero, the symmetric and alternating squares are subrepresentations of the second tensor power. They can be used to define the Frobenius–Schur indicator, which indicates whether a given irreducible character is real, complex, or quaternionic. They are examples of Schur functors. They are defined as follows.

One can define a linear transformation on any vector space with a basis by defining its action on basis elements and extending linearly. Let V be a vector space with basis ${\displaystyle \{v_{1},v_{2},\ldots ,v_{n}\}}$. The tensor product ${\displaystyle V\otimes V}$ has a basis ${\displaystyle \{v_{i}\otimes v_{j}\mid i,j\in \{1,2,\ldots ,n\}\}}$. Define an endomorphism (self-map) T of ${\displaystyle V\otimes V}$ as follows:

{\displaystyle {\begin{aligned}T:V\otimes V&\longrightarrow V\otimes V\\v_{i}\otimes v_{j}&\longmapsto v_{j}\otimes v_{i}\end{aligned}}}

This transformation is independent of the chosen basis. It is an involution (it is its own inverse), and so is an automorphism (self-isomorphism) of ${\displaystyle V\otimes V}$.

Define two subsets of the second tensor power of V:

{\displaystyle {\begin{aligned}\operatorname {Sym} ^{2}(V)&:=\{v\in V\otimes V\mid T(v)=v\}\\\operatorname {Alt} ^{2}(V)&:=\{v\in V\otimes V\mid T(v)=-v\}\end{aligned}}}

These are the symmetric square of V and the alternating square of V, respectively.[9] The symmetric and alternating squares are also known as the symmetric part and antisymmetric part of the tensor product.[10]

Properties

The second tensor power of a linear representation V of a group G decomposes as the direct sum of the symmetric and alternating squares:

${\displaystyle V^{\otimes 2}=V\otimes V\cong \operatorname {Sym} ^{2}(V)\oplus \operatorname {Alt} ^{2}(V)}$

as representations. In particular, both are subrepresenations of the second tensor power. In the language of modules over the group ring, the symmetric and alternating squares are ${\displaystyle \mathbb {C} [G]}$-submodules of ${\displaystyle V\otimes V}$.[11]

If V has a basis ${\displaystyle \{v_{1},v_{2},\ldots ,v_{n}\}}$, then the symmetric square has a basis ${\displaystyle \{v_{i}\otimes v_{j}+v_{j}\otimes v_{i}\mid 1\leq i\leq j\leq n\}}$ and the alternating square has a basis ${\displaystyle \{v_{i}\otimes v_{j}-v_{j}\otimes v_{i}\mid 1\leq i. Accordingly,

{\displaystyle {\begin{aligned}\dim \operatorname {Sym} ^{2}(V)&={\frac {\dim V(\dim V+1)}{2}},\\\dim \operatorname {Alt} ^{2}(V)&={\frac {\dim V(\dim V-1)}{2}}.\end{aligned}}}[12][9]

Let ${\displaystyle \chi :G\to \mathbb {C} }$ be the character of ${\displaystyle V}$. Then we can calculate the characters of the symmetric and alternating squares as follows: for all g in G,

{\displaystyle {\begin{aligned}\chi _{\operatorname {Sym} ^{2}(V)}(g)&={\frac {1}{2}}(\chi (g)^{2}+\chi (g^{2})),\\\chi _{\operatorname {Alt} ^{2}(V)}(g)&={\frac {1}{2}}(\chi (g)^{2}-\chi (g^{2})).\end{aligned}}}[13]

The symmetric and exterior powers

As in multilinear algebra, over a field of characteristic zero, one can more generally define the kth symmetric power ${\displaystyle \operatorname {Sym} ^{n}(V)}$ and kth exterior power ${\displaystyle \Lambda ^{n}(V)}$, which are subspaces of the kth tensor power (see those pages for more detail on this construction). They are also subrepresentations, but higher tensor powers no longer decompose as their direct sum.

The Schur–Weyl duality computes the irreducible representations occurring in tensor powers of representations of the general linear group ${\displaystyle G=\operatorname {GL} (V)}$. Precisely, as an ${\displaystyle S_{n}\times G}$-module

${\displaystyle V^{\otimes n}\simeq \bigoplus _{\lambda }M_{\lambda }\otimes S^{\lambda }(V)}$

where

• ${\displaystyle M_{\lambda }}$ is an irreducible representation of the symmetric group ${\displaystyle S_{n}}$ corresponding to a partition ${\displaystyle \lambda }$ of n (in decreasing order),
• ${\displaystyle S^{\lambda }(V)}$ is the image of the Young symmetrizer ${\displaystyle c_{\lambda }:V^{\otimes n}\to V^{\otimes n}}$.

The mapping ${\displaystyle V\mapsto S^{\lambda }(V)}$ is a functor called the Schur functor. It generalizes the constructions of symmetric and exterior powers:

${\displaystyle S^{(n)}(V)=\operatorname {Sym} ^{n}V,\,\,S^{(1,1,\dots ,1)}(V)=\wedge ^{n}V.}$

In particular, as an G-module, the above simplifies to

${\displaystyle V^{\otimes n}\simeq \bigoplus _{\lambda }S^{\lambda }(V)^{\oplus m_{\lambda }}}$

where ${\displaystyle m_{\lambda }=\dim M_{\lambda }}$. Moreover, the multiplicity ${\displaystyle m_{\lambda }}$ may be computed by the Frobenius formula (or the hook length formula). For example, take ${\displaystyle n=3}$. Then there are exactly three partitions: ${\displaystyle 3=3=2+1=1+1+1}$ and, as it turns out, ${\displaystyle m_{(3)}=m_{(1,1,1)}=1,\,m_{(2,1)}=2}$. Hence,

${\displaystyle V^{\otimes 3}\simeq \operatorname {Sym} ^{3}V\bigoplus \wedge ^{3}V\bigoplus S^{(2,1)}(V)^{\oplus 2}.}$

Tensor products involving Schur functors

Let ${\displaystyle S^{\lambda }}$ denote the Schur functor defined according to a partition ${\displaystyle \lambda }$. Then there is the following decomposition:[14]

${\displaystyle S^{\lambda }V\otimes S^{\mu }V\simeq \bigoplus _{\nu }(S^{\nu }V)^{\oplus N_{\lambda \mu \nu }}}$

where the multiplicities ${\displaystyle N_{\lambda \mu \nu }}$ are given by the Littlewood–Richardson rule.

Given finite-dimensional vector spaces V, W, the Schur functors Sλ give the decomposition

${\displaystyle \operatorname {Sym} (W^{*}\otimes V)\simeq \bigoplus _{\lambda }S^{\lambda }(W^{*})\otimes S^{\lambda }(V)}$

The left-hand side can be identified with the ring k[Hom(V, W)] = k[V *W] of polynomial functions on Hom(V, W) and so the above also gives the decomposition of k[Hom(V, W)].

Tensor products representations as representations of product groups

Let G, H be two groups and let ${\displaystyle (\pi ,V)}$ and ${\displaystyle (\rho ,W)}$ be representations of G and H, respectively. Then we can let the direct product group ${\displaystyle G\times H}$ act on the tensor product space ${\displaystyle V\otimes W}$ by the formula

${\displaystyle (g,h)\cdot (v\otimes w)=\pi (g)v\otimes \rho (h)w.}$

Even if ${\displaystyle G=H}$, we can still perform this construction, so that the tensor product of two representations of ${\displaystyle G}$ could, alternatively, be viewed as a representation of ${\displaystyle G\times G}$ rather than a representation of ${\displaystyle G}$. It is therefore important to clarify whether the tensor product of two representations of ${\displaystyle G}$ is being viewed as a representation of ${\displaystyle G}$ or as a representation of ${\displaystyle G\times G}$.

In contrast to the Clebsch–Gordan problem discussed above, the tensor product of two irreducible representations of ${\displaystyle G}$ is irreducible when viewed as a representation of the product group ${\displaystyle G\times G}$.

Notes

1. Serre 1977, p. 8.
2. Fulton & Harris 1991, p. 4.
3. Hall 2015 Section 4.3.2
4. Hall 2015 Definition 4.19
5. Hall 2015 Proposition 4.18
6. Hall 2015 pp. 433–434
7. Hall 2015 Theorem C.1
8. Hall 2015 Proof of Proposition 6.17
9. Serre 1977, p. 9.
10. James 2001, p. 196.
11. James 2001, Proposition 19.12.
12. James 2001, Proposition 19.13.
13. James 2001, Proposition 19.14.
14. Fulton–Harris, § 6.1. just after Corollay 6.6.

References

• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666.
• James, Gordon Douglas (2001). Representations and characters of groups. Liebeck, Martin W. (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 978-0521003926. OCLC 52220683.
• Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402 .
• Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. Springer-Verlag. ISBN 978-0-387-90190-9. OCLC 2202385.