Tensor product of 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.

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 the map ${\displaystyle \pi _{1}\otimes \pi _{2}:{\mathfrak {g}}\rightarrow \operatorname {End} (V_{1}\otimes V_{2})}$ given by[4]

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

where ${\displaystyle I}$ 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 ${\displaystyle \pi _{1}}$ 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]

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 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 ${\displaystyle \ell }$, 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 ${\displaystyle 4\times 3=6+4+2}$.

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 ${\displaystyle U\otimes V}$.

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 (self-map) T of ${\displaystyle V\otimes V}$ as follows:

${\displaystyle {\begin{aligned}T:V\otimes V&\longrightarrow V\otimes V\\v\otimes w&\longmapsto w\otimes v.\end{aligned}}}$[9]

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, ${\displaystyle V\odot V}$, and the alternating square of V, ${\displaystyle V\wedge V}$, respectively.[10] The symmetric and alternating squares are also known as the symmetric part and antisymmetric part of the tensor product.[11]

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 subrepresentations 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 ${\displaystyle V\otimes V}$.[12]

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 ${\displaystyle \{v_{i}\otimes v_{j}-v_{j}\otimes v_{i}\mid 1\leq i<j\leq n\}}$. Accordingly,

${\displaystyle {\begin{aligned}\dim \operatorname {Sym} ^{2}(V)&={\frac {\dim V(\dim V+1)}{2}},\\\dim \operatorname {Alt} ^{2}(V)&={\frac {\dim V(\dim V-1)}{2}}.\end{aligned}}}$[13][10]

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,

${\displaystyle {\begin{aligned}\chi _{\operatorname {Sym} ^{2}(V)}(g)&={\frac {1}{2}}(\chi (g)^{2}+\chi (g^{2})),\\\chi _{\operatorname {Alt} ^{2}(V)}(g)&={\frac {1}{2}}(\chi (g)^{2}-\chi (g^{2})).\end{aligned}}}$[14]

As in multilinear algebra, over a field of characteristic zero, one can more generally define the k^th symmetric power ${\displaystyle \operatorname {Sym} ^{n}(V)}$ and k^th exterior power ${\displaystyle \Lambda ^{n}(V)}$, 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 ${\displaystyle G=\operatorname {GL} (V)}$. 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,

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