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 characteristic of K does not divide the order of G. Let φ1,…,φn be a complete set (up to equivalence) of distinct irreducible
(http://planetmath.org/GroupRepresentation) (linear) representations of G over K, so that φi is a homomorphism
:
φi:G⟶GL(ni,K) |
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:G→K is defined by
χi(g)=Trace(φi(g)). |
Notice, however, that in general the map χi is not a homomorphism from the group into either the additive or multiplicative group of K. We define a system of primitive orthogonal idempotents of the group ring
K[G], one for each χi, by:
𝟏χi=1|G|∑g∈Gχi(g-1)g∈K[G] |
so that ∑i𝟏χi=1∈K and 𝟏χi⋅𝟏χj=δij where δij is the Kronecker delta function. We define the χi component of
K[G]to be the ideal K[G]χi=𝟏χi⋅K[G]. Notice that Vi=K[G]χi is a finite dimensional K-vector space
, 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 dimension
ni. In particular, there is a decomposition:
K[G]=⊕χK[G]χ. |
If k∈K[G] then by the previous decomposition, we can write:
k=∑χkχ |
where kχ∈K[G]χ. Notice that the representations φi can be retrieved as:
φi:G⟶GL(K[G]χi). |
Lemma.
Let M be a K[G]-module and define submodules Mχ=1χ⋅M, for each irreducible character χ. Then:
-
1.
There is a decomposition M=⊕χMχ.
-
2.
The group K[G] acts on Mχ via K[G]χ. In other words, if k∈K[G], with k=∑χkχ then:
k⋅m=kχ⋅m, for all m∈Mχ. -
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.
φ=φi⊕…⊕φi=φ⊕ri for some integer r≥0. In other words, Mχi is the submodule consisting of the sum of all K[G]-submodules of M isomorphic to K[G]χi.
-
4.
Suppose that M, N and R are K[G]-modules which fit in the short exact sequence
:
0⟶R⟶M⟶N⟶0 where every map above is a K[G]-module homomorphism
, i.e. each map is a K-homomorphism which is compatible with the action of G. Then, the exact sequence
above yields an exact sequence of χ components:
0⟶Rχ⟶Mχ⟶Nχ⟶0 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 |