decomposition of a module using orthogonal idempotents

Let K be a field and let G be a finite abelian group. For simplicity, we will assume that the characteristicPlanetmathPlanetmath of K does not divide the order of G. Let φ1,,φn be a complete set (up to equivalence) of distinct irreduciblePlanetmathPlanetmath ( (linear) representations of G over K, so that φi is a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:


where ni is the degree of the representation φi and ini=|G|. Let χ1,,χn be the irreducible characters attached to the φi, i.e. the function χi:GK is defined by


Notice, however, that in general the map χi is not a homomorphism from the group into either the additive or multiplicative groupMathworldPlanetmath of K. We define a system of primitive orthogonal idempotents of the group ringMathworldPlanetmath K[G], one for each χi, by:


so that i𝟏χi=1K and 𝟏χi𝟏χj=δij where δij is the Kronecker delta function. We define the χi componentMathworldPlanetmathPlanetmath of K[G]to be the ideal K[G]χi=𝟏χiK[G]. Notice that Vi=K[G]χi is a finite dimensional K-vector spaceMathworldPlanetmath, on which G acts. Thus, the representation of G afforded by the K[G]-module Vi, call it φ, must be one of the representations φj defined above. Comparing the trace, one concludes that φ=φi and Vi=K[G]χi is a vector space of dimensionPlanetmathPlanetmath ni. In particular, there is a decomposition:


If kK[G] then by the previous decomposition, we can write:


where kχK[G]χ. Notice that the representations φi can be retrieved as:


Let M be a K[G]-module and define submodulesMathworldPlanetmath Mχ=1χM, for each irreducible character χ. Then:

  1. 1.

    There is a decomposition M=χMχ.

  2. 2.

    The group K[G] acts on Mχ via K[G]χ. In other words, if kK[G], with k=χkχ then:

    km=kχm, for all mMχ.
  3. 3.

    The representation φ of G afforded by the K-vector space Mχi is, up to equivalence, a number of copies of φi, i.e.


    for some integer r0. In other words, Mχi is the submodule consisting of the sum of all K[G]-submodules of M isomorphic to K[G]χi.

  4. 4.

    Suppose that M, N and R are K[G]-modules which fit in the short exact sequenceMathworldPlanetmathPlanetmath:


    where every map above is a K[G]-module homomorphismMathworldPlanetmath, i.e. each map is a K-homomorphism which is compatible with the action of G. Then, the exact sequencePlanetmathPlanetmathPlanetmathPlanetmath above yields an exact sequence of χ components:


    for every irreducible character χ.

Title decomposition of a module using orthogonal idempotents
Canonical name DecompositionOfAModuleUsingOrthogonalIdempotents
Date of creation 2013-03-22 15:12:22
Last modified on 2013-03-22 15:12:22
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 9
Author alozano (2414)
Entry type Application
Classification msc 13C05
Classification msc 16S34