At 9:00, I don't understand how you went from average of the values of the character of Hom(V,W) to dim Hom_G (V,W). I see that the lemma gives that the expression in the second line is the dim of G-linear maps that are fixed by G. Why is this equal to dim Hom_G(V,W). Doesn't Hom_G(V,W) contain maps that commute with each rho_g? ie, rho_g (phi) = phi (rho_g) for all g in G. But maps fixed by G only satisfy rho_g(phi) = phi. Why are these the same?
@Kan : The expression "rho_g(phi)" has two different meaning in the end of your post. The first one refers to the action of G on W, the second one to the action of G on Hom(V,W). The point is that Hom_G(V,W) is equal to Hom(V,W)^G, in other words the equivariant linear maps from V to W are exactly the linear maps from V to W that are fixed by the action of G on Hom(V,W). Hope this helps.
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At 9:00, I don't understand how you went from average of the values of the character of Hom(V,W) to dim Hom_G (V,W). I see that the lemma gives that the expression in the second line is the dim of G-linear maps that are fixed by G. Why is this equal to dim Hom_G(V,W). Doesn't Hom_G(V,W) contain maps that commute with each rho_g? ie, rho_g (phi) = phi (rho_g) for all g in G. But maps fixed by G only satisfy rho_g(phi) = phi. Why are these the same?
There is a small step missing here, one can make Hom(V,W) to Hom_G(V,W) by mapping \phi to 1/|G|\sum_g g\phi.
@Kan : The expression "rho_g(phi)" has two different meaning in the end of your post. The first one refers to the action of G on W, the second one to the action of G on Hom(V,W). The point is that Hom_G(V,W) is equal to Hom(V,W)^G, in other words the equivariant linear maps from V to W are exactly the linear maps from V to W that are fixed by the action of G on Hom(V,W).
Hope this helps.
At 5:55, 1. in the definition of hermitian inner product, there is an inconsistency in notation. = \bar{a} + \bar{b} .
Shouldn't you assume right from the beginning (or recall that hypothesis) that the group G is finite ?