Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $V = e\mathbb{C}G$ be a submodule of $\mathbb{C}G$ and $e \in \mathbb{C}G$ is such that $e^{2} = e$. Why is it that $Hom_{\mathbb{C}G}(V, V) \cong e\mathbb{C}Ge$?

share|improve this question
add comment

1 Answer 1

up vote 2 down vote accepted

There is a natural map $\alpha: e \mathbb{C}[G] e \rightarrow \mathrm{End}_G(V)$ defined by $$\alpha(efe)(h)=efeh \quad \hbox{for $f \in \mathbb{C}[G]$ and $h \in e \mathbb{C}[G]$.}$$

Using the fact that a $G$-endomorphism $\phi:V \rightarrow V$ is determined by the image of $e$ it is straightforward to check that $\alpha$ is an isomorphism.

share|improve this answer
    
Why is $\alpha$ surjective? Is this where the idempotency of $e$ comes in? –  109238 Mar 1 '12 at 0:29
    
Yes: given a $G$-endo $\phi$ of $V$, using $e^2=e$, we obtain that $\phi(e)$ is an element of $V$ satisfying $\phi(e)=\phi(e)e \in Ve$. Thus $\phi(e)=efe$ for some $f \in \mathbb{C} [G]$ and one checks that $\alpha(efe)=\phi$. –  S123 Mar 1 '12 at 0:33
    
@Steve: I'm wondering: If I understand correctly, $\mathbb CG$ and $e\mathbb CG$ are being considered as right modules here. Am I missing a notational convention that indicated that in the question, or did you just infer that from the content? I was under the impression that $\mathbb CG$ is usually considered as a left module unless stated otherwise. –  joriki Mar 1 '12 at 7:43
    
Joriki, well there are those, mostly dwelling in shadows and shunned by their neighbors, who actually prefer right modules to left modules. In this case, the beginning "Let $V=e\mathbb{C}[G]$ be a submodules..." of the question suggests that we are working with right modules, since the alternative is having to type an annoying inverse everywhere. –  S123 Mar 1 '12 at 12:22
    
:-) Some day the neighbours will see the light... By the way, I happened to check back on this; if you want me to get notified of your response, you have to put an @ in front of the user name. –  joriki Mar 1 '12 at 15:40
show 1 more comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.