The Tensor Product and the Tensor Product of Representations
Point of post: In this post we discuss the notion of the tensor product and the tensor product of two representations.
I have previously discussed the notion of the tensor product, but will be considering a slightly different (although isomorphic) construction. In general any vector space with a blinear map such that if and are bases for and then can be considered a tensor product in some sense, since this generalized construction defines a unique, up to isomorphism structure. That’s the view we’ll take here with a special focus on the particular case of treating as which is the space of all anti-bilinear forms. We then define the tensor product of two representations.
Bilinear Maps and the Tensor Product
We’ve previously discussed the notion of bilinear forms as being maps which are linear in each entry (where are some -spaces) in the sense that for each fixed the map is linear as is the map for each fixed . We can easily extend this notion of “bilinearity” to more general cases. In particular, if are -spaces then a map is called a bilinear map (or just bilinear) if the maps and are linear for each fixed and .
Let be -spaces and a distinguished bilinear map such that if and are bases for and then is a basis for . We then call a tensor product of and under and denote this . It’s clear that for any tensor product of and the dimension of the tensor product is and thus all tensor products of and are isomorphic.
Anti-Bilinear Forms As A Tensor Product
Let and be pre-Hilbert spaces with inner products and . We define the space of anti-bilinear forms, denoted , to be the set of all maps such that is antilinear in each entry. This is clearly a vector space with pointwise addition and scalar multiplication of anti-bilinear forms. Our current goal is to show that we can think of as a tensor product with distinguished bilinear map
Of course this is equivalent to showing that is bilinear map which carries bases to bases. But before we do this we make a quick observation in the form of the following theorem
Theorem: Let and be pre-Hilbert spaces (really it’s true even for spaces without specified inner products, but this will suffice) then is dimensional.
Proof: It’s easily verified, using the same techniques as in the theory of the space of all bilinear forms that if is a basis for and a basis for then
is a basis for where is the unique anti-bilinear form such that .
With this in mind we go on to prove that can be viewed as a tensor product. Since the fact that the map is evidently bilinear it suffices to prove that carries bases to bases. But, this is true. Indeed:
Theorem: Let be defined as above, then if is a basis for and a basis for then is a basis for .
Proof: By our previous theorem it suffices to show that is linearly independent, since . Indeed, suppose that
Then, fix and . Note that since there exists some non-zero . We see then that otherwise would be perpendicular to a basis and thus zero, contrary to construction. Similarly, we may choose such that . It follows then that
and since and are non-zero we may conclude that . Since and was arbitrary the conclusion follows.
It follows then that can be considered as an actual tensor product. From now on out we shall denote as (when discussing the topic of rep. theory) and shall always denote as it did in the above.
We now wish to endow with an inner product structure so we can start discussing the tensor product of representations. We begin with a theorem
Theorem: There exists a unique inner product on such that .
Proof: While clear since this defines it on a basis we show a proof nonetheless. Suppose that were such an inner product and and bases for and . Then, we’d then see that the following would necessarily be true:
and it’s evident from this formulation that the right hand side of the above is, in fact, an inner product. Thus, it remains to show that this inner product satisfies the desired quality. To do this we let , and . Then,
But, with equal validity
from where the conclusion follows.
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.