# Abstract Nonsense

## The Tensor Product and the Tensor Product of Representations (Pt. II)

Point of post: This is a continuation of this post.

It’s fairly easy to see from this that with this inner product $\otimes$ not only carries bases to bases but orthonormal bases to orthonormal bases.

We now wish to define the tensor product of endomorphisms on $\mathscr{V}$ and $\mathscr{W}$. Indeed, if $A\in\text{End}\left(\mathscr{V}\right)$ and $B\in\text{End}\left(\mathscr{W}\right)$ we define the tensor product of $A$ and $B$ to be the map

$A\otimes B:\mathscr{V}\otimes\mathscr{W}\to\mathscr{V}\to\mathscr{W}$

where $\left((A\otimes B)(C)\right)(x,y)=C\left(A^\ast (x),B^\ast (y)\right)$. This is indeed a linear transformation since

\begin{aligned}\left(\left(A\otimes B\right)\left(\alpha C+\beta D\right)\right)(x,y) &=\left(\alpha C+\beta D\right)\left(A^\ast(x),B^\ast(y)\right)\\ &=\alpha C\left(A^{\ast}(x),B^\ast(y)\right)+\beta D\left(A^{\ast}(x),B^{\ast}(y)\right)\\ &= \alpha \left(\left(A\otimes B\right)(C)\right)(x,y)+\beta\left(\left(A\otimes B\right)\left(D\right)\right)(x,y)\end{aligned}

Note that $A\otimes B$ is defined precisely such that $\left(A\otimes B\right)(v\otimes w)=\left(A(v)\right)\otimes\left(B(w)\right)$. Indeed,

\begin{aligned}\left(\left(A\otimes B\right)\left(v\otimes w\right)\right)(x,y) &=\left(v\otimes w\right)\left(A^{\ast}(x),B^{\ast}(y)\right)\\ &=\left\langle v,A^{\ast}(x)\right\rangle_1\left\langle w,B^{\ast}(y)\right\rangle_2\\ &= \left\langle A(v),x\right\rangle_1\left\langle B(w),y\right\rangle_2\\ &= \left(\left(A(v)\right)\otimes B(w)\right)(x,y)\end{aligned}

We claim that if $A\in\mathcal{U}\left(\mathscr{V}\right)$ and $B\in\mathcal{U}\left(\mathscr{W}\right)$ then $A\otimes B\in\mathcal{U}\left(\mathscr{V}\otimes\mathscr{W}\right)$. Indeed, since $\left\{v\otimes w:v\in\mathscr{V}\text{ and }w\in\mathscr{W}\right\}$ contains a basis for $\mathscr{V}\otimes\mathscr{W}$ it suffices to check that

$\left\langle \left(A\otimes B\right)\left(u\otimes v\right),\left(A\otimes B\right)\left(e\otimes f\right)\right\rangle=\left\langle u\otimes v,e\otimes f\right\rangle$

But, this is fairly clear since

\begin{aligned}\left\langle \left(A\otimes B\right)\left(u\otimes v\right),\left(A\otimes B\right)\left(e\otimes f\right)\right\rangle &=\left\langle \left(\left(A\otimes B\right)\left(u\otimes v\right)\right),\left(A(e)\right)\otimes\left(B(f)\right)\right\rangle\\ &=\left(A(u)\otimes B(v)\right)\left(A(e),B(f)\right)\\ &=\left(u\otimes v\right)\left(A^\ast A(e),B^\ast B(f)\right)\\ &= \left(u\otimes v\right)\left(e,f\right)\\ &= \left\langle u\otimes v,e\otimes f\right\rangle\end{aligned}

It thus makes sense then to define the tensor product of the representations $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ and $\rho':G\to\mathcal{U}\left(\mathscr{V}\right)$ to be the map $\rho\otimes\rho':G\to\mathcal{U}\left(\mathscr{V}\otimes\mathscr{W}\right)$ by $g\mapsto \rho_g\otimes \rho'_g$. The only thing to check is that $\rho\otimes\rho'$ is a homomorphism, but this is clear. Indeed,

\begin{aligned}\left(\left(\rho\otimes \rho'\right)(g)\left(\rho\otimes\rho'\right)(g')\right)(u\otimes v) &= \left(\rho\otimes \rho'\right)\left(\left(\rho(g')(u)\right)\otimes\left(\rho'(g')(v)\right)\right)\\ &= \left(\rho(g)\rho(g')(u)\right)\otimes\left(\rho'(g)\rho'(g')(v)\right)\\ &=\left(\rho(gg')(u)\right)\otimes\left(\rho'(gg')(v)\right)\\ &=\left(\left(\rho\otimes \rho'\right)(gg')\right)\left(u\otimes v\right)\end{aligned}

and thus since they agree on a basis we see that $\left(\rho\otimes \rho'\right)(g)\left(\rho\otimes\rho'\right)(g')=\left(\rho\otimes\rho'\right)(gg')$ and the conclusion follows.

Unfortunately one’s hopes would be that if $\rho$ and $\rho'$ are irreps that $\rho\otimes\rho'$ is also an irrep, but this is not true. But, by first principles we know that

$\displaystyle \rho\otimes\rho'\cong \bigoplus_{j=1}^{m}\rho_j$

for some irreps $\rho_1,\cdots,\rho_m$

References:

1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print.