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orthogonal idempotents of the group ring (Definition)

Let $ G$ be a finite abelian group, let $ L$ be any field containing the $ \vert G\vert$-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

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

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)$.



"orthogonal idempotents of the group ring" is owned by mathcam.
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decomposition of a module using orthogonal idempotents (Application) by alozano
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Cross-references: modules, orthogonal idempotents, equalities, group, character, roots of unity, field, abelian group, finite
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This is version 6 of orthogonal idempotents of the group ring, born on 2004-02-27, modified 2005-05-15.
Object id is 5646, canonical name is OrthogonalIdempotentsOfTheGroupRing.
Accessed 1568 times total.

Classification:
AMS MSC: 16S34 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Group rings , Laurent polynomial rings)
Pending Errata and Addenda
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