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:

$${\phi }_i:G⟶\mathrm{GL}(n_i,K)$$

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:

$${\mathrm{𝟏}}_{{\chi }_i}=\frac{1}{|G|}\sum _{g\in G}{\chi }_i(g^{-1})g\in K[G]$$

so that ${\sum }_i{\mathrm{𝟏}}_{{\chi }_i}=1\in K$ and ${\mathrm{𝟏}}_{{\chi }_i}\cdot {\mathrm{𝟏}}_{\chi j}={\delta }_{ij}$ where ${\delta }_{ij}$ is the Kronecker delta function. We define the ${\chi }_i$ component of$K[G]$to be the ideal $K{[G]}_{{\chi }_i}={\mathrm{𝟏}}_{{\chi }_i}\cdot K[G]$. 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:

$${\phi }_i:G⟶\mathrm{GL}(\mathrm{K}{[\mathrm{G}]}_{{\chi }_{\mathrm{i}}}).$$

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:

   $$0⟶R⟶M⟶N⟶0$$

   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:

   $$0⟶R_{\chi }⟶M_{\chi }⟶N_{\chi }⟶0$$

   for every irreducible character $\chi$.