Basic Representation Theory I'm reading a paper that uses representation theory, and I'm stuck on something simple. Say $\Delta$ is an abelian group, and $\hat{\Delta}$ its group of irreducible characters. Say $M$ is a $\mathbb{C}[\Delta]$-module. The paper uses the decomposition $M=\oplus_{\chi \in \hat{\Delta}} M^{\chi}$, where $M^{\chi}:=e_{\chi}M$ and $e_{\chi}:=\frac{1}{|\Delta|} \sum_{\delta \in \Delta} \chi(\delta)\delta^{-1}$.
The question, roughly speaking, is: how should I think of the $M^{\chi}$'s? What is their meaning, and function? How else are they characterized?
I've taken a course in representation theory a few years back, so it's all a little vague, but if I remember correctly, $\mathbb{C}[\Delta]$ is the direct sum of $|\Delta|$ many simple submodules, in bijection with the $\chi$'s. Then the $e_{\chi}$'s can be interpreted as being the decomposition of $1$ ($1=\sum_{\chi} e_{\chi}$, where $e_{\chi}$ is in the simple submodule of $\mathbb{C}[\Delta]$ corresponding to $\chi$).
But I don't see how this translates to a decomposition of some random module $M$. $M$ is a direct sum of simple submodules. But there's reason to think that there are $|\Delta|$ many (for example: what if $M$ is simple?). So what is the interpretation of the $M^{\chi}$'s?
 A: The $M^{\chi}$ are called isotypic components in the literature. You should think of $M^{\chi}$ as the subspace of $M$ consisting of elements which "transform like $\chi$" under the action of $\Delta$. 
This will make much more sense to you once you master a simple example. Consider the group $\mathbb{Z}/2\mathbb{Z}$ acting on, say, the space of continuous functions $\mathbb{R} \to \mathbb{R}$ via $f(x) \mapsto f(-x)$. This space decomposes into two isotypic components: the even functions and the odd functions (corresponding to the trivial and nontrivial character of $\mathbb{Z}/2\mathbb{Z}$). Any function is uniquely a sum of an even and an odd function since
$$f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}.$$
The statement you are asking about is just a natural generalization of this idea. 
A: The category of representations of a finite group  over $\mathbb{C}$ (or more generally a field whose characteristic is prime to the order of the group) is a semisimple category: every object splits as a direct sum of finitely many simple objects. In fact, one can directly show that, if the group $G$ is abelian, then $\mathbb{C}[G]$ is isomorphic to $G$ copies of $\mathbb{C}$ as a ring; to do this, note that $\mathbb{C}[G]$ can be identified with complex-valued functions on $G$ with convolution the product. Now the Fourier transform induces an isomorphism of complex-valued functions on $G$ with complex-valued functions on $G$, that takes convolution to multiplication.
So if we think of $\mathbb{C}[G]$ as a commutative ring, it is isomorphic (via the Fourier transform) to the ring of functions on $G$ with the multiplication being pointwise multiplication. In other words, this is just the direct product ring $\prod_G \mathbb{C}$. 
What are the modules over $\prod_G \mathbb{C}$? Well, any such splits as a finite direct sum of modules over which the $i$th factor acts by multiplication by complex numbers, and where the other factors act by zero. In general, a module over a product ring $R_1 \times R_2$ decomposes uniquely as a direct sum of an $R_1$-module (on which $R_2$ acts trivially) and an $R_2$-module (on which $R_1$ acts trivially) simply because we have idempotents $e_1, e_2$ in the product ring, and can take their images.
So what does this correspond to if we have a $\mathbb{C}[G]$-module and want to decompose it? Well, the above construction shows that we need to find the primitive idempotents. Since the Fourier transform gave us an isomorphism with $\mathbb{C}^G$, and the latter has an obvious set of idempotents, we see that there is a correspondence of primitive idempotents. This gives that the idempotents in $\mathbb{C}[G]$ correspond precisely to the characters (since the characters define the Fourier transform) and in the way you describe.
So the decomposition is just a special instance of the general fact alluded to above about modules over a direct product of rings. (This works even when $G$ is non-abelian, though then the group algebra decomposes as a product of matrix algebras, by general structure theory for semisimple algebras.)
A: Although I'm sure Akhil is aware of this, for other readers it might be best to clafiry:
Finite-dimensional $\mathbb{C}[G]$-modules are always semi-simple, as is the case when $\mathbb{C}$ is replaced by any other field of characteristic $0$. The category of representations over a field of prime characteristic, however, is not semi-simple. Any group whose order is divisible by the characteristic of the field will have a a representation which does not decompose into irreducible representations.
