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[parent] decomposition of a module using orthogonal idempotents (Application)

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 (linear) representations of $ G$ over $ K$, so that $ \varphi_i$ is a homomorphism:

$\displaystyle \varphi_i\colon G \longrightarrow \operatorname{GL}(n_i,K)$
where $ n_i$ is the degree of the representation $ \varphi_i$ and $ \sum_i n_i=\vert G\vert$. 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
$\displaystyle \chi_i(g)=$Trace$\displaystyle (\varphi_i(g)).$
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:
$\displaystyle {\bf 1}_{\chi_i}=\frac{1}{\vert G\vert}\sum_{g\in G} \chi_i(g^{-1})g\in K[G]$
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:
$\displaystyle K[G]=\oplus_\chi K[G]_\chi.$
If $ k\in K[G]$ then by the previous decomposition, we can write:
$\displaystyle k=\sum_\chi k_\chi$
where $ k_\chi \in K[G]_\chi$. Notice that the representations $ \varphi_i$ can be retrieved as:
$\displaystyle \varphi_i\colon G \longrightarrow \operatorname{GL(K[G]_{\chi_i})}.$
Lemma 1   Let $ M$ be a $ K[G]$-module and define submodules $ M_\chi={\bf 1}_{\chi}\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:
    $\displaystyle k\cdot m = k_\chi \cdot m,$    for all $\displaystyle m\in M_\chi.$
  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.
    $\displaystyle \varphi=\varphi_i\oplus \ldots \oplus \varphi_i=\varphi_i^{\oplus r}$
    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$ isomorphic to $ K[G]_{\chi_i}$.
  4. Suppose that $ M$, $ N$ and $ R$ are $ K[G]$-modules which fit in the short exact sequence:
    $\displaystyle 0\longrightarrow R \longrightarrow M \longrightarrow N \longrightarrow 0$
    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:
    $\displaystyle 0\longrightarrow R_\chi \longrightarrow M_\chi \longrightarrow N_\chi \longrightarrow 0$
    for every irreducible character $ \chi$.



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Cross-references: exact sequence, action, compatible, short exact sequence, isomorphic, sum, integer, acts on, submodules, decomposition, dimension, vector space, trace, finite dimensional, ideal, component, Kronecker delta, orthogonal idempotents of the group ring, primitive, multiplicative group, additive, group, map, function, characters, irreducible, degree, representations, equivalence, complete, order, divide, characteristic, abelian group, finite, field

This is version 6 of decomposition of a module using orthogonal idempotents, born on 2005-04-25, modified 2005-04-27.
Object id is 6966, canonical name is DecompositionOfAModuleUsingOrthogonalIdempotents.
Accessed 858 times total.

Classification:
AMS MSC: 16S34 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Group rings , Laurent polynomial rings)
 13C05 (Commutative rings and algebras :: Theory of modules and ideals :: Structure, classification theorems)
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