I'm trying to solve an exercise from MIT open course in representations(p.9):
I'm trying to understend:
- If there is motivation for defining such $e_\chi$. It looks like something very explicit, but I didn't find it in any other place.
- How to solve it. I tried to build such an isomorphism by hand, and didn't manage to. Is there an important useful theorem that I missed or something?
** ** Edit ** **
I think I found an approach for building the isomorphism, I'm still missing some details, and I'll be happy for some help:
We have $Ind^K_G \mathbb C _\chi = \mathbb C [G] \underset {\mathbb C [K]} {\otimes} \mathbb C _\chi$
Now we can define a function $$\varphi : \mathbb C [G] \underset {\mathbb C [K]} {\otimes} \mathbb C _\chi \rightarrow \mathbb C [G] e_\chi$$ $$ g\otimes c \mapsto cge_\chi$$ It is quite natural to build it. I'm not sure how to show that $\varphi$ is well-defined, but I don't think it should be hard, and it is quite easy to see it's surjective.
To show it's injective I tried to look at it as a function between vector spaces, and show that the vector spaces have the same dimension. The domain is clearly from dimension $[G:K]$. How do I prove that the image has the same dimension?
The motivation for $e_\chi$ is: up to a scalar, that's what it has to be so that the span of $e_\chi$ is a submodule of $\mathbb{C}K$ isomorphic to $\mathbb{C}_\chi$. The choice of scalar makes it idempotent.
Here are two suggestions for showing $\mathbb{C}Ge_\chi$ is isomorphic to the induced module. The first is to verify that it satisfies the universal property of an induced module $X$:
$$ \begin{array}{lll} \mathbb{C}_\chi &\stackrel {\forall \psi} \to & M \\ &\stackrel\phi\searrow & \uparrow \exists! T \\ & & X \end{array} $$
"There is a $\mathbb{C}K$ homomorphism $\mathbb{C}_\chi \to X|_K$ such that for every $\mathbb{C}G$-module $M$ and every $\mathbb{C}K$-homomorphism $\mathbb{C}_\chi \to M|_K$ there is a unique $\mathbb{C}G$ homomorphism $T : X \to M$ making the diagram commute."
The second is to write down a homomorphism from the usual construction $\mathbb{C}G \otimes _K \mathbb{C}_\chi$ to $\mathbb{C}G e_\chi$. There is an obvious choice, but you'll need to show it is well-defined ($(gk) \otimes 1$ and $g \otimes ( k \cdot 1)$ have the same image) using the explicit definition of $e_\chi$. You will then want to know it is an isomorphism - you could show that it is surjective, for example, and then show the two modules have the same dimension using explicit bases for each. Obviously the elements $ge_\chi, g \in G$ span, but they aren't linearly independent. But if you restrict $g$ to a set of coset reps...