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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{1},V_{2}} are linear representations of a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} , then their tensor product is the tensor product of vector spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{1}\otimes V_{2}} with the linear action of $G$ uniquely determined by the condition that

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

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

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

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

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

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 $T(V)$ is an algebraic representation of G; i.e., each element of G acts as an algebra automorphism.

Lie algebra representations

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

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

where $I$ is the identity endomorphism. This is called the Kronecker sum, defined in Matrix addition#Kronecker sum and Kronecker product#Properties. The motivation for the use of the Kronecker sum in this definition comes from the case in which $\pi _{1}$ and $\pi _{2}$ come from representations $\Pi _{1}$ and $\Pi _{2}$ of a Lie group $G$. In that case, a simple computation shows that the Lie algebra representation associated to $\Pi _{1}\otimes \Pi _{2}$ is given by the preceding formula.

Quantum groups

For quantum groups, the coproduct is no longer co-commutative. As a result, the natural permutation map $V\otimes W\rightarrow W\otimes V$ is no longer an isomorphism of modules. However, the permutation map remains an isomorphism of vector spaces.

Action on linear maps

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

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

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

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

as vector spaces; this vector space isomorphism is in fact an isomorphism of representations.

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

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

Let $E=\operatorname {End} (V)$ denote the endomorphism algebra of V and let A denote the subalgebra of $E^{\otimes m}$ 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 $V_{1},V_{2}$ of a group or Lie algebra is usually not irreducible. It is therefore of interest to attempt to decompose $V_{1}\otimes V_{2}$ into irreducible pieces. This decomposition problem is known as the Clebsch–Gordan problem.

The SU(2) case

Further information: Clebsch–Gordan coefficients

The prototypical 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 $\ell$, whose possible values are

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

(The dimension of the representation is then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\ell +1} .) Let us take two parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell \geq m} . Then the tensor product representation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{\ell }\otimes V_{m}} then decomposes as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{3/2}} and the three-dimensional representation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{1}} . The tensor product representation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{3/2}\otimes V_{1}} has dimension 12 and decomposes as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\times 3=6+4+2} .

The SU(3) case

Main article: Clebsch–Gordan coefficients for SU(3)

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (m_{1},m_{2})} , one takes the tensor product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{1}} copies of the standard representation and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

In contrast to the situation for SU(2), in the Clebsch–Gordan decomposition for SU(3), a given irreducible representation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} may occur more than once in the decomposition of $U\otimes V$.

Tensor power

As with vector spaces, one can define the k^th tensor power of a representation V to be the vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Let V be a vector space. Define an endomorphism T of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\otimes V} as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\begin{aligned}T:V\otimes V&\longrightarrow V\otimes V\\v\otimes w&\longmapsto w\otimes v.\end{aligned}}}

It is an involution (its own inverse), and so is an automorphism of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\otimes V} .

Define two subsets of the second tensor power of V,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\odot V} , and the alternating square of V, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\wedge V} , respectively. The symmetric and alternating squares are also known as the symmetric part and antisymmetric part of the tensor product.

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{\otimes 2}=V\otimes V\cong \operatorname {Sym} ^{2}(V)\oplus \operatorname {Alt} ^{2}(V)}

as representations. In particular, both are subrepresentations of the second tensor power. In the language of modules over the group ring, the symmetric and alternating squares are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb {C} [G]} -submodules of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\otimes V} .

If V has a basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{1},v_{2},\ldots ,v_{n}\}} , then the symmetric square has a basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{i}\otimes v_{j}-v_{j}\otimes v_{i}\mid 1\leq i<j\leq n\}} . Accordingly,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}}

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi :G\to \mathbb {C} } be the character of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . Then we can calculate the characters of the symmetric and alternating squares as follows: for all g in G,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}}

The symmetric and exterior powers

As in multilinear algebra, over a field of characteristic zero, one can more generally define the k^th symmetric power Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname {Sym} ^{n}(V)} and k^th exterior power Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda ^{n}(V)} , which are subspaces of the k^th 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G=\operatorname {GL} (V)} . Precisely, as an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{n}\times G} -module

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{\otimes n}\simeq \bigoplus _{\lambda }M_{\lambda }\otimes S^{\lambda }(V)}

where

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{\lambda }} is an irreducible representation of the symmetric group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm {S} _{n}} corresponding to a partition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } of n (in decreasing order),
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{\lambda }(V)} is the image of the Young symmetrizer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{\lambda }:V^{\otimes n}\to V^{\otimes n}} .

The mapping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\mapsto S^{\lambda }(V)} is a functor called the Schur functor. It generalizes the constructions of symmetric and exterior powers:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{(n)}(V)=\operatorname {Sym} ^{n}V,\,\,S^{(1,1,\dots ,1)}(V)=\wedge ^{n}V.}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{\otimes n}\simeq \bigoplus _{\lambda }S^{\lambda }(V)^{\oplus m_{\lambda }}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{\lambda }=\dim M_{\lambda }} . Moreover, the multiplicity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{\lambda }} may be computed by the Frobenius formula (or the hook length formula). For example, take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=3} . Then there are exactly three partitions: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3=3=2+1=1+1+1} and, as it turns out, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{(3)}=m_{(1,1,1)}=1,\,m_{(2,1)}=2} . Hence,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{\lambda }} denote the Schur functor defined according to a partition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } . Then there is the following decomposition:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{\lambda }V\otimes S^{\mu }V\simeq \bigoplus _{\nu }(S^{\nu }V)^{\oplus N_{\lambda \mu \nu }}}

where the multiplicities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{\lambda \mu \nu }} are given by the Littlewood–Richardson rule.

Given finite-dimensional vector spaces V, W, the Schur functors S^λ give the decomposition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 of polynomial functions on Hom(V, W ), k[Hom(V, W )] = k[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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\pi ,V)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\rho ,W)} be representations of G and H, respectively. Then we can let the direct product group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\times H} act on the tensor product space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\otimes W} by the formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g,h)\cdot (v\otimes w)=\pi (g)v\otimes \rho (h)w.}

Even if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G=H} , we can still perform this construction, so that the tensor product of two representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} could, alternatively, be viewed as a representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\times G} rather than a representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} . It is therefore important to clarify whether the tensor product of two representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is being viewed as a representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} or as a representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\times G} .

In contrast to the Clebsch–Gordan problem discussed above, the tensor product of two irreducible representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is irreducible when viewed as a representation of the product group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\times G} .

See also

• Dual representation
• Hermite reciprocity
• Clebsch–Gordan coefficients
• Lie group representation
• Lie algebra representation
• Kronecker product
• Hopf algebra

References

• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 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, vol. 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.