decomposition of a module using orthogonal idempotents
Let $K$ be a field and let $G$ be a finite abelian group. For simplicity, we will assume that the characteristic^{} of $K$ does not divide the order of $G$. Let ${\phi}_{1},\mathrm{\dots},{\phi}_{n}$ be a complete set (up to equivalence) of distinct irreducible^{} (http://planetmath.org/GroupRepresentation) (linear) representations of $G$ over $K$, so that ${\phi}_{i}$ is a homomorphism^{}:
$${\phi}_{i}:G\u27f6\mathrm{GL}({n}_{i},K)$$ 
where ${n}_{i}$ is the degree of the representation ${\phi}_{i}$ and ${\sum}_{i}{n}_{i}=G$. Let ${\chi}_{1},\mathrm{\dots},{\chi}_{n}$ be the irreducible characters attached to the ${\phi}_{i}$, i.e. the function ${\chi}_{i}:G\to K$ is defined by
$${\chi}_{i}(g)=\text{Trace}({\phi}_{i}(g)).$$ 
Notice, however, that in general the map ${\chi}_{i}$ is not a homomorphism from the group into either the additive or multiplicative group^{} of $K$. We define a system of primitive orthogonal idempotents of the group ring^{} $K[G]$, one for each ${\chi}_{i}$, by:
$${\mathrm{\U0001d7cf}}_{{\chi}_{i}}=\frac{1}{G}\sum _{g\in G}{\chi}_{i}({g}^{1})g\in K[G]$$ 
so that ${\sum}_{i}{\mathrm{\U0001d7cf}}_{{\chi}_{i}}=1\in K$ and ${\mathrm{\U0001d7cf}}_{{\chi}_{i}}\cdot {\mathrm{\U0001d7cf}}_{\chi j}={\delta}_{ij}$ where ${\delta}_{ij}$ is the Kronecker delta function. We define the ${\chi}_{i}$ component^{} of $K[G]$to be the ideal $K{[G]}_{{\chi}_{i}}={\mathrm{\U0001d7cf}}_{{\chi}_{i}}\cdot K[G]$. Notice that ${V}_{i}=K{[G]}_{{\chi}_{i}}$ is a finite dimensional $K$vector space^{}, on which $G$ acts. Thus, the representation of $G$ afforded by the $K[G]$module ${V}_{i}$, call it $\phi $, must be one of the representations ${\phi}_{j}$ defined above. Comparing the trace, one concludes that $\phi ={\phi}_{i}$ and ${V}_{i}=K{[G]}_{{\chi}_{i}}$ is a vector space of dimension^{} ${n}_{i}$. In particular, there is a decomposition:
$$K[G]={\oplus}_{\chi}K{[G]}_{\chi}.$$ 
If $k\in K[G]$ then by the previous decomposition, we can write:
$$k=\sum _{\chi}{k}_{\chi}$$ 
where ${k}_{\chi}\in K{[G]}_{\chi}$. Notice that the representations ${\phi}_{i}$ can be retrieved as:
$${\phi}_{i}:G\u27f6\mathrm{GL}(\mathrm{K}{[\mathrm{G}]}_{{\chi}_{\mathrm{i}}}).$$ 
Lemma.
Let $M$ be a $K\mathit{}\mathrm{[}G\mathrm{]}$module and define submodules^{} ${M}_{\chi}\mathrm{=}{\mathrm{1}}_{\chi}\mathrm{\cdot}M$, for each irreducible character $\chi $. Then:

1.
There is a decomposition $M={\oplus}_{\chi}{M}_{\chi}$.

2.
The group $K[G]$ acts on ${M}_{\chi}$ via $K{[G]}_{\chi}$. In other words, if $k\in K[G]$, with $k={\sum}_{\chi}{k}_{\chi}$ then:
$$k\cdot m={k}_{\chi}\cdot m,\mathit{\text{for all}}m\in {M}_{\chi}.$$ 
3.
The representation $\phi $ of $G$ afforded by the $K$vector space ${M}_{{\chi}_{i}}$ is, up to equivalence, a number of copies of ${\phi}_{i}$, i.e.
$$\phi ={\phi}_{i}\oplus \mathrm{\dots}\oplus {\phi}_{i}={\phi}_{i}^{\oplus r}$$ for some integer $r\ge 0$. In other words, ${M}_{{\chi}_{i}}$ is the submodule consisting of the sum of all $K[G]$submodules of $M$ isomorphic to $K{[G]}_{{\chi}_{i}}$.

4.
Suppose that $M$, $N$ and $R$ are $K[G]$modules which fit in the short exact sequence^{}:
$$0\u27f6R\u27f6M\u27f6N\u27f60$$ where every map above is a $K[G]$module homomorphism^{}, i.e. each map is a $K$homomorphism which is compatible with the action of $G$. Then, the exact sequence^{} above yields an exact sequence of $\chi $ components:
$$0\u27f6{R}_{\chi}\u27f6{M}_{\chi}\u27f6{N}_{\chi}\u27f60$$ for every irreducible character $\chi $.
Title  decomposition of a module using orthogonal idempotents 

Canonical name  DecompositionOfAModuleUsingOrthogonalIdempotents 
Date of creation  20130322 15:12:22 
Last modified on  20130322 15:12:22 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  9 
Author  alozano (2414) 
Entry type  Application 
Classification  msc 13C05 
Classification  msc 16S34 