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:

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

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

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

• ${\epsilon }_{\chi }\cdot g=\chi (g){\epsilon }_{\chi }$

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