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 
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 
![\left[\sigma(a)\right]\left(f(v)\right)=f\left(\left[\rho(a)\right](v)\right)](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28f%28v%29%5Cright%29%3Df%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29&bg=e6e6e6&fg=333333&s=0&c=20201002)
![\displaystyle\begin{aligned}\left[\sigma(a)\right]\left(\left[cf\right](v)\right)=\left[\sigma(a)\right]\left(cf(v)\right)\\=c\left[\sigma(a)\right]\left(f(v)\right)\\=cf\left(\left[\rho(a)\right](v)\right)\\=\left[cf\right]\left(\left[\rho(a)\right](v)\right)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28%5Cleft%5Bcf%5Cright%5D%28v%29%5Cright%29%3D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28cf%28v%29%5Cright%29%5C%5C%3Dc%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28f%28v%29%5Cright%29%5C%5C%3Dcf%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29%5C%5C%3D%5Cleft%5Bcf%5Cright%5D%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
And so
Now if 
![\displaystyle\begin{aligned}\left[\sigma(a)\right]\left(\left[f+g\right](v)\right)=\left[\sigma(a)\right]\left(f(v)+g(v)\right)\\=\left[\sigma(a)\right]\left(f(v)\right)+\left[\sigma(a)\right]\left(g(v)\right)\\=f\left(\left[\rho(a)\right](v)\right)+g\left(\left[\rho(a)\right](v)\right)\\=\left[f+g\right]\left(\left[\rho(a)\right](v)\right)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28%5Cleft%5Bf%2Bg%5Cright%5D%28v%29%5Cright%29%3D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28f%28v%29%2Bg%28v%29%5Cright%29%5C%5C%3D%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28f%28v%29%5Cright%29%2B%5Cleft%5B%5Csigma%28a%29%5Cright%5D%5Cleft%28g%28v%29%5Cright%29%5C%5C%3Df%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29%2Bg%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29%5C%5C%3D%5Cleft%5Bf%2Bg%5Cright%5D%5Cleft%28%5Cleft%5B%5Crho%28a%29%5Cright%5D%28v%29%5Cright%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
Which shows that
Since composition of intertwiners is the same as composing their linear maps, it’s also bilinear. It immediately follows that the category 
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[…] 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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