# 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 $\hat{G}$ denote the character group of $G$ with values in $L$. For any character $\chi\in\hat{G}$, we define $\varepsilon_{\chi}$, the corresponding orthogonal idempotent of the group ring $L[G]$, by

 $\displaystyle\varepsilon_{\chi}=\frac{1}{|G|}\sum_{g\in G}\chi(g)g^{-1}.$

The following equalities hold:

• $\varepsilon_{\chi}^{2}=\varepsilon_{\chi}$ for all $\chi$

• $\varepsilon_{\chi}\varepsilon_{\psi}=0$ for any $\chi\neq\psi$

• $\sum_{\chi\in\hat{G}}\varepsilon_{\chi}=1$

• $\varepsilon_{\chi}\cdot g=\chi(g)\varepsilon_{\chi}$

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

Title orthogonal idempotents of the group ring OrthogonalIdempotentsOfTheGroupRing 2013-03-22 14:12:42 2013-03-22 14:12:42 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 16S34