Abstract Nonsense

Crushing one theorem at a time

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.

Advertisements

February 14, 2011 - Posted by | Algebra, Representation Theory | , ,

No comments yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: