orthogonal idempotents of the group ring
Let $G$ be a finite abelian group, let $L$ be any field containing the $G$th roots of unity^{}, and let $\widehat{G}$ denote the character group of $G$ with values in $L$. For any character^{} $\chi \in \widehat{G}$, we define ${\epsilon}_{\chi}$, the corresponding orthogonal idempotent of the group ring^{} $L[G]$, by
${\epsilon}_{\chi}={\displaystyle \frac{1}{G}}{\displaystyle \sum _{g\in G}}\chi (g){g}^{1}.$ 
The following equalities hold:

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${\epsilon}_{\chi}^{2}={\epsilon}_{\chi}$ for all $\chi $

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${\epsilon}_{\chi}{\epsilon}_{\psi}=0$ for any $\chi \ne \psi $

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${\sum}_{\chi \in \widehat{G}}{\epsilon}_{\chi}=1$

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${\epsilon}_{\chi}\cdot g=\chi (g){\epsilon}_{\chi}$
These orthogonal idempotents are used to decompose modules over $L[G]$: If $M$ is such a module, then $M={\oplus}_{\chi}({\epsilon}_{\chi}M)$.
Title  orthogonal idempotents of the group ring 

Canonical name  OrthogonalIdempotentsOfTheGroupRing 
Date of creation  20130322 14:12:42 
Last modified on  20130322 14:12:42 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  9 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 16S34 