## Intertwiner Spaces

Another grading day, another straightforward post. It should come as no surprise that the collection of intertwining maps between any two representations forms a vector space.

Let’s fix representations and . We already know that is a vector space. We also know that an intertwiner can be identified with a linear map . What I’m asserting is that is actually a *subspace* of under this identification.

Indeed, all we really need to check is that this subset is closed under additions and under scalar multiplications. For the latter, let’s say that is an intertwiner. That is, . Then given a constant we consider the linear map we calculate

And so is an intertwiner as well.

Now if and are both intertwiners, satisfying conditions like the one above, we consider their sum and calculate

Which shows that is again an intertwiner.

Since composition of intertwiners is the same as composing their linear maps, it’s also bilinear. It immediately follows that the category is enriched over .

[...] of all, the intertwiners between any two representations form a vector space, which is really an abelian group plus stuff. Since the composition of intertwiners is bilinear, [...]

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