orthogonal idempotents of the group ring

Let G be a finite abelian group, let L be any field containing the |G|-th roots of unityMathworldPlanetmath, and let G^ denote the character group of G with values in L. For any characterPlanetmathPlanetmath χG^, we define εχ, the corresponding orthogonal idempotent of the group ringMathworldPlanetmath L[G], by


The following equalities hold:

  • εχ2=εχ for all χ

  • εχεψ=0 for any χψ

  • χG^εχ=1

  • εχg=χ(g)εχ

These orthogonal idempotents are used to decompose modules over L[G]: If M is such a module, then M=χ(εχM).

Title orthogonal idempotents of the group ring
Canonical name OrthogonalIdempotentsOfTheGroupRing
Date of creation 2013-03-22 14:12:42
Last modified on 2013-03-22 14:12:42
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 16S34