# 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 $\varphi_{1},\ldots,\varphi_{n}$ be a complete set (up to equivalence) of distinct irreducible (http://planetmath.org/GroupRepresentation) (linear) representations of $G$ over $K$, so that $\varphi_{i}$ is a homomorphism:

 $\varphi_{i}\colon G\longrightarrow\operatorname{GL}(n_{i},K)$

where $n_{i}$ is the degree of the representation $\varphi_{i}$ and $\sum_{i}n_{i}=|G|$. Let $\chi_{1},\ldots,\chi_{n}$ be the irreducible characters attached to the $\varphi_{i}$, i.e. the function $\chi_{i}\colon G\to K$ is defined by

 $\chi_{i}(g)=\text{Trace}(\varphi_{i}(g)).$

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:

 ${\bf 1}_{\chi_{i}}=\frac{1}{|G|}\sum_{g\in G}\chi_{i}(g^{-1})g\in K[G]$

so that $\sum_{i}{\bf 1}_{\chi_{i}}=1\in K$ and ${\bf 1}_{\chi_{i}}\cdot{\bf 1}_{\chi j}=\delta_{ij}$ where $\delta_{ij}$ is the Kronecker delta function. We define the $\chi_{i}$ component of $K[G]$to be the ideal $K[G]_{\chi_{i}}={\bf 1}_{\chi_{i}}\cdot K[G]$. 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 $\varphi$, must be one of the representations $\varphi_{j}$ defined above. Comparing the trace, one concludes that $\varphi=\varphi_{i}$ 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 $\varphi_{i}$ can be retrieved as:

 $\varphi_{i}\colon G\longrightarrow\operatorname{GL(K[G]_{\chi_{i}})}.$
###### Lemma.

Let $M$ be a $K[G]$-module and define submodules $M_{\chi}={\bf 1}_{\chi}\cdot M$, for each irreducible character $\chi$. Then:

1. 1.

There is a decomposition $M=\oplus_{\chi}M_{\chi}$.

2. 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:

 $k\cdot m=k_{\chi}\cdot m,\text{ for all }m\in M_{\chi}.$
3. 3.

The representation $\varphi$ of $G$ afforded by the $K$-vector space $M_{\chi_{i}}$ is, up to equivalence, a number of copies of $\varphi_{i}$, i.e.

 $\varphi=\varphi_{i}\oplus\ldots\oplus\varphi_{i}=\varphi_{i}^{\oplus r}$

for some integer $r\geq 0$. In other words, $M_{\chi_{i}}$ is the submodule consisting of the sum of all $K[G]$-submodules of $M$$K[G]_{\chi_{i}}$.

4. 4.

Suppose that $M$, $N$ and $R$ are $K[G]$-modules which fit in the short exact sequence:

 $0\longrightarrow R\longrightarrow M\longrightarrow N\longrightarrow 0$

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:

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

for every irreducible character $\chi$.

Title decomposition of a module using orthogonal idempotents DecompositionOfAModuleUsingOrthogonalIdempotents 2013-03-22 15:12:22 2013-03-22 15:12:22 alozano (2414) alozano (2414) 9 alozano (2414) Application msc 13C05 msc 16S34