## 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 not only carries bases to bases but orthonormal bases to orthonormal bases.

We now wish to define the *tensor product *of endomorphisms on and . Indeed, if and we define the *tensor product *of and to be the map

where . This is indeed a linear transformation since

Note that is defined precisely such that . Indeed,

We claim that if and then . Indeed, since contains a basis for it suffices to check that

But, this is fairly clear since

It thus makes sense then to define the *tensor product *of the representations and to be the map by . The only thing to check is that is a homomorphism, but this is clear. Indeed,

and thus since they agree on a basis we see that and the conclusion follows.

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

for some irreps

**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.

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