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In [[Mathematics|mathematics]], the '''tensor product of representations''' is a tensor product of vector spaces underlying [[Representation theory|representations]] together with the factor-wise [[Group action|group action]] on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional [[Irreducible representation|irreducible representation]]s if one already knows a few.
== Definition ==
===Group representations===
If <math>V_1, V_2</math> are linear representations of a group <math>G</math>, then their tensor product is the tensor product of [[Vector space|vector space]]s <math>V_1 \otimes V_2</math> with the linear action of <math>G</math> uniquely determined by the condition that
:<math>g \cdot (v_1 \otimes v_2) = (g\cdot v_1) \otimes (g\cdot v_2)</math>{{sfn|Serre|1977|p=8}}{{sfn|Fulton|Harris|1991|p=4}}
for all <math>v_1\in V_1</math> and <math>v_2\in V_2</math>. Although not every element of <math>V_1 \otimes V_2</math> is expressible in the form <math>v_1\otimes v_2</math>, the [[Tensor product#Universal property|universal property of the tensor product]] guarantees that this action is well-defined.

In the language of [[Group homomorphism|homomorphisms]], if the actions of <math>G</math> on <math>V_1</math> and <math>V_2</math> are given by homomorphisms <math>\Pi_1: G\to\operatorname{GL}(V_1)</math> and <math>\Pi_2: G\to\operatorname{GL}(V_2)</math>, then the tensor product representation is given by the homomorphism <math>\Pi_1\otimes\Pi_2: G\to\operatorname{GL}(V_1 \otimes V_2)</math> given by
:<math>\Pi_1\otimes\Pi_2(g) = \Pi_1(g)\otimes\Pi_2(g)</math>,
where <math>\Pi_1(g)\otimes\Pi_2(g)</math> is the [[Tensor product#Tensor product of linear maps|tensor product of linear maps]].<ref>{{harvnb|Hall|2015}} Section 4.3.2</ref>

One can extend the notion of tensor products to any finite number of representations. If ''V'' is a linear representation of a [[Group (mathematics)|group]] ''G'', then with the above linear action, the [[Tensor algebra|tensor algebra]] <math>T(V)</math> is an [[Algebraic representation|algebraic representation]] of ''G''; i.e., each element of ''G'' acts as an algebra automorphism.

===Lie algebra representations===
If <math>(V_1,\pi_1)</math> and <math>(V_2,\pi_2)</math> are representations of a [[Lie algebra]] <math>\mathfrak g</math>, then the tensor product of these representations is the map <math>\pi_1\otimes\pi_2:\mathfrak g\to\operatorname{End}(V_1 \otimes V_2)</math> given by<ref>{{harvnb|Hall|2015}} Definition 4.19</ref>
:<math>\pi_1\otimes\pi_2(X)=\pi_1(X)\otimes I+I\otimes\pi_2(X)</math>, 
where <math>I</math> is the [[Identity function|identity]] [[Endomorphism|endomorphism]]. This is called the Kronecker sum, defined in [[Matrix addition]] and [[Kronecker product]].
The motivation for the use of the Kronecker sum in this definition comes from the case in which <math>\pi_1</math> and <math>\pi_2</math> come from representations <math>\Pi_1</math> and <math>\Pi_2</math> of a [[Lie group]] <math>G</math>. In that case, a simple computation shows that the Lie algebra representation associated to <math>\Pi_1\otimes\Pi_2</math> is given by the preceding formula.<ref>{{harvnb|Hall|2015}} Proposition 4.18</ref>

== Action on linear maps ==
If <math>(V_1, \Pi_1)</math> and <math>(V_2, \Pi_2)</math> are representations of a group <math>G</math>, let <math>\operatorname{Hom}(V_1,V_2)</math> denote the [[Vector space|space]] of all [[Linear map|linear map]]s from <math>V_1</math> to <math>V_2</math>. Then <math>\operatorname{Hom}(V_1,V_2)</math> can be given the structure of a representation by defining
:<math>g\cdot A=\Pi_2(g)A\Pi_1(g)^{-1}</math>
for all <math>A\in\operatorname{Hom}(V,W)</math>. Now, there is [[Tensor product#Tensor product vs. Hom|a natural isomorphism]]
:<math>\operatorname{Hom}(V, W)\cong V^* \otimes W</math>
as vector spaces;{{sfn|Fulton|Harris|1991|p=4}} this vector space [[Isomorphism|isomorphism]] is in fact an isomorphism of representations.<ref>{{harvnb|Hall|2015}} pp. 433–434</ref>

The trivial subrepresentation <math>\operatorname{Hom}(V, W)^G</math> consists of [[Equivariant map|''G''-linear maps]]; i.e.,
:<math>\operatorname{Hom}_G(V, W) = \operatorname{Hom}(V, W)^G.</math>

Let <math>E = \operatorname{End}(V)</math> denote the endomorphism algebra of ''V'' and let ''A'' denote the subalgebra of <math>E^{\otimes m}</math> consisting of symmetric tensors. The main theorem of invariant theory states that ''A'' is [[Semisimple algebra|semisimple]] when the [[Characteristic (algebra)|characteristic]] of the base [[Field (mathematics)|field]] is zero.

==Clebsch–Gordan theory==
===The general problem===
The tensor product of two irreducible representations <math>V_1,V_2</math> of a group or Lie algebra is usually not irreducible. It is therefore of interest to attempt to decompose <math>V_1\otimes V_2</math> into irreducible pieces. This decomposition problem is known as the Clebsch–Gordan problem.

===The SU(2) case===
{{further|Physics:Clebsch–Gordan coefficients}}
The [[Physics:Clebsch–Gordan coefficients|prototypical example]] of this problem is the case of the [[Rotation group SO(3)|rotation group SO(3)]]—or its double cover, the [[Special unitary group#The group SU(2)|special unitary group SU(2)]]. The irreducible representations of SU(2) are described by a parameter <math>\ell</math>, whose possible values are
:<math>\ell=0, 1/2, 1, 3/2, \ldots.</math>
(The dimension of the representation is then <math>2\ell+1</math>.) Let us take two parameters <math>\ell</math> and <math>m</math> with <math>\ell\geq m</math>. Then the tensor product representation <math>V_\ell\otimes V_m</math> then decomposes as follows:<ref>{{harvnb|Hall|2015}} Theorem C.1</ref>
:<math>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}.</math>
Consider, as an example, the tensor product of the four-dimensional representation <math>V_{3/2}</math> and the three-dimensional representation <math>V_1</math>. The tensor product representation <math>V_{3/2}\otimes V_1</math> has dimension 12 and decomposes as
:<math>V_{3/2}\otimes V_1\cong V_{5/2}\oplus V_{3/2}\oplus V_{1/2}</math>,
where the representations on the right-hand side have dimension 6, 4, and 2, respectively. We may summarize this result arithmetically as <math>4 \times 3 = 6+4+2</math>.

===The SU(3) case===
{{main|Physics:Clebsch–Gordan coefficients for SU(3)}}
In the case of the group SU(3), all [[Lie algebra representation#The case of sl(3,C)|the irreducible representations]] can be generated from the standard 3-dimensional representation and its dual, as follows. To generate the representation with label <math>(m_1,m_2)</math>, one takes the tensor product of <math>m_1</math> copies of the standard representation and <math>m_2</math> copies of the dual of the standard representation, and then takes the invariant subspace generated by the tensor product of the highest weight vectors.<ref>{{harvnb|Hall|2015}} Proof of Proposition 6.17</ref>

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

== Tensor power ==

As with vector spaces, one can define the '''{{var|k}}<sup>th</sup> tensor power of a representation''' {{var|V}} to be the vector space <math>V^{\otimes k}</math> with the action given above.

=== The symmetric and alternating square ===

Over a field of characteristic zero, the symmetric and alternating squares are [[Subrepresentation|subrepresentation]]s of the second tensor power. They can be used to define the [[Frobenius–Schur indicator]], which indicates whether a given irreducible character is [[Real representation|real]], [[Complex representation|complex]], or [[Quaternionic representation|quaternionic]]. They are examples of [[Schur functor]]s.
They are defined as follows.

Let {{var|V}} be a vector space. Define an endomorphism {{var|T}} of <math>V \otimes V</math> as follows:
:<math>\begin{align}
  T: V \otimes V &\longrightarrow V \otimes V \\
  v \otimes w &\longmapsto w \otimes v.
\end{align}</math><ref>Precisely, we have <math>V \times V \to V \otimes V, (v, w) \mapsto v \otimes w</math>, which is bilinear and thus descends to the linear map <math>V \otimes V \to V \otimes V.</math></ref>

It is an [[Involution (mathematics)|involution]] (its own inverse), and so is an [[Automorphism|automorphism]] of <math>V \otimes V</math>.

Define two subsets of the second tensor power of {{var|V}},
:<math display="block">\begin{align}
  \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{align}</math>

These are the '''symmetric square of {{var|V}}''', <math>V \odot V</math>,  and the '''alternating square of {{var|V}}''', <math>V\wedge V </math>, respectively.{{sfn|Serre|1977|p=9}} The symmetric and alternating squares are also known as the '''symmetric part''' and '''antisymmetric part''' of the tensor product.{{sfn|James|2001|p=196}}

==== Properties ====

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

: <math display="block">V^{\otimes 2} = V \otimes V \cong \operatorname{Sym}^2(V) \oplus \operatorname{Alt}^2(V)</math>

as representations. In particular, both are [[Subrepresentation|subrepresentation]]s of the second tensor power. In the language of [[Module (mathematics)|module]]s over the [[Group ring|group ring]], the symmetric and alternating squares are <math>\mathbb{C}[G]</math>-submodules of <math>V \otimes V</math>.{{sfn|James|2001|loc=Proposition 19.12}}

If {{var|V}} has a [[Basis (linear algebra)|basis]] <math>\{v_1,v_2,\ldots,v_n\}</math>, then the symmetric square has a basis <math>\{v_i\otimes v_j+v_j\otimes v_i\mid 1\leq i\leq j\leq n\}</math> and the alternating square has a basis <math>\{v_i\otimes v_j-v_j\otimes v_i\mid 1\leq i<j\leq n\}</math>. Accordingly,

:<math>\begin{align}
  \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{align}</math>{{sfn|James|2001|loc=Proposition 19.13}}{{sfn|Serre|1977|p=9}}

Let <math>\chi:G\to\mathbb{C}</math> be the character of <math>V</math>. Then we can calculate the characters of the symmetric and alternating squares as follows: for all {{var|g}} in {{var|G}},
:<math>\begin{align}
  \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{align}</math>{{sfn|James|2001|loc=Proposition 19.14}}

=== The symmetric and exterior powers ===

As in [[Multilinear algebra|multilinear algebra]], over a field of characteristic zero, one can more generally define the {{var|k}}<sup>th</sup> [[Symmetric power|symmetric power]] <math>\operatorname{Sym}^n(V)</math> and {{var|k}}<sup>th</sup> exterior power <math>\Lambda^n(V)</math>, which are [[Linear subspace|subspaces]] of the {{var|k}}<sup>th</sup> 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|general linear group]] <math>G = \operatorname{GL}(V)</math>. Precisely, as an <math>S_n \times G</math>-module
:<math>V^{\otimes n} \simeq \bigoplus_\lambda M_{\lambda} \otimes S^{\lambda}(V)</math>
where
*<math>M_{\lambda}</math> is an irreducible representation of the [[Symmetric group|symmetric group]] <math>\mathrm{S}_n</math> corresponding to a partition <math>\lambda</math> of ''n'' (in decreasing order),
*<math>S^{\lambda}(V)</math> is the image of the [[Young symmetrizer]] <math>c_{\lambda}: V^{\otimes n} \to V^{\otimes n}</math>.

The mapping <math>V \mapsto S^{\lambda}(V)</math> is a [[Functor|functor]] called the [[Schur functor]]. It generalizes the constructions of symmetric and exterior powers:
:<math>S^{(n)}(V) = \operatorname{Sym}^n V, \,\, S^{(1, 1, \dots, 1)}(V) = \wedge^n V.</math>

In particular, as a ''G''-module, the above simplifies to
:<math>V^{\otimes n} \simeq \bigoplus_{\lambda} S^{\lambda}(V)^{\oplus m_{\lambda}}</math>
where <math>m_\lambda = \dim M_\lambda</math>. Moreover, the multiplicity <math>m_{\lambda}</math> may be computed by the [[Frobenius formula]] (or the [[Hook length formula|hook length formula]]). For example, take <math>n = 3</math>. Then there are exactly three partitions: <math>3 = 3 = 2 + 1 = 1 + 1 + 1</math> and, as it turns out, <math>m_{(3)} = m_{(1, 1, 1)} = 1, \, m_{(2, 1)} = 2</math>. Hence,
:<math>V^{\otimes 3} \simeq \operatorname{Sym}^3 V \bigoplus \wedge^3 V \bigoplus S^{(2, 1)}(V)^{\oplus 2}.</math>

== Tensor products involving Schur functors ==
Let <math>S^{\lambda}</math> denote the [[Schur functor]] defined according to a partition <math>\lambda</math>. Then there is the following decomposition:<ref>{{harvnb|Fulton–Harris|loc=§ 6.1. just after Corollay 6.6.}}</ref>
:<math>S^{\lambda} V \otimes S^{\mu} V \simeq \bigoplus_{\nu} (S^{\nu} V)^{\oplus N_{\lambda \mu \nu}}</math>
where the multiplicities <math>N_{\lambda \mu \nu}</math> are given by the [[Littlewood–Richardson rule]].

Given finite-dimensional vector spaces ''V'', ''W'', the [[Schur functor]]s ''S''<sup>λ</sup> give the decomposition
:<math>\operatorname{Sym}(W^* \otimes V) \simeq \bigoplus_{\lambda} S^{\lambda}(W^*) \otimes S^{\lambda}(V)</math>
The left-hand side can be identified with the [[Ring of polynomial functions|ring of polynomial functions]] on Hom(''V'',&thinsp;''W''&hairsp;), ''k''[Hom(''V'',&thinsp;''W''&hairsp;)] = ''k''[''V''<sup>&thinsp;*</sup>&thinsp;⊗&thinsp;''W''&hairsp;], and so the above also gives the decomposition of ''k''[Hom(''V'',&thinsp;''W''&hairsp;)].

== Tensor products representations as representations of product groups ==
Let ''G'', ''H'' be two groups and let <math>(\pi,V)</math> and <math>(\rho,W)</math> be representations of ''G'' and ''H'', respectively. Then we can let the [[Direct product of groups|direct product group]] <math>G \times H</math> act on the tensor product space <math>V \otimes W</math> by the formula
:<math>(g, h) \cdot (v \otimes w) = \pi(g) v \otimes \rho(h) w.</math>
Even if <math>G=H</math>, we can still perform this construction, so that the tensor product of two representations of <math>G</math> could, alternatively, be viewed as a representation of <math>G \times G</math> rather than a representation of <math>G</math>. It is therefore important to clarify whether the tensor product of two representations of <math>G</math> is being viewed as a representation of <math>G</math> or as a representation of <math>G \times G</math>.

In contrast to the Clebsch–Gordan problem discussed above, the tensor product of two irreducible representations of <math>G</math> is irreducible when viewed as a representation of the product group <math>G \times G</math>.

== See also ==
*[[Dual representation]]
*[[Hermite reciprocity]]
*[[Physics:Clebsch–Gordan coefficients|Clebsch–Gordan coefficients]]
*Lie group representation
*[[Lie algebra representation]]
*[[Kronecker product]]
*[[Hopf algebra]]

== Notes ==
{{reflist}}

== References ==
*{{Fulton-Harris}}
* {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition= 2nd|series=Graduate Texts in Mathematics|volume=222 |publisher=Springer|year=2015|isbn=978-3319134666}}.
* {{Cite book|title=Representations and characters of groups|last=James|first=Gordon Douglas|date=2001|publisher=Cambridge University Press|others=Liebeck, Martin W.|isbn=978-0521003926|edition=2nd|location=Cambridge, UK|oclc=52220683}}
*[[Biography:Claudio Procesi|Claudio Procesi]] (2007) ''Lie Groups: an approach through invariants and representation'', Springer, {{isbn|9780387260402}} .
* {{cite book|first=Jean-Pierre|last=Serre|title=Linear Representations of Finite Groups|url=https://archive.org/details/linearrepresenta1977serr|url-access=registration|publisher=Springer-Verlag|year=1977|isbn=978-0-387-90190-9|oclc=2202385}}

[[Category:Representation theory of finite groups]]

{{Sourceattribution|Tensor product of representations}}