decomposition of a module using orthogonal idempotents
Let be a field and let be a finite abelian group. For simplicity, we will assume that the characteristic of does not divide the order of . Let be a complete set (up to equivalence) of distinct irreducible (http://planetmath.org/GroupRepresentation) (linear) representations of over , so that is a homomorphism:
where is the degree of the representation and . Let be the irreducible characters attached to the , i.e. the function is defined by
Notice, however, that in general the map is not a homomorphism from the group into either the additive or multiplicative group of . We define a system of primitive orthogonal idempotents of the group ring , one for each , by:
so that and where is the Kronecker delta function. We define the component of to be the ideal . Notice that is a finite dimensional -vector space, on which acts. Thus, the representation of afforded by the -module , call it , must be one of the representations defined above. Comparing the trace, one concludes that and is a vector space of dimension . In particular, there is a decomposition:
If then by the previous decomposition, we can write:
where . Notice that the representations can be retrieved as:
Let be a -module and define submodules , for each irreducible character . Then:
There is a decomposition .
The group acts on via . In other words, if , with then:
The representation of afforded by the -vector space is, up to equivalence, a number of copies of , i.e.
for some integer . In other words, is the submodule consisting of the sum of all -submodules of isomorphic to .
|Title||decomposition of a module using orthogonal idempotents|
|Date of creation||2013-03-22 15:12:22|
|Last modified on||2013-03-22 15:12:22|
|Last modified by||alozano (2414)|