[GAP Forum] working with GroupRings

[Thomas Breuer]

Dear Forum,

concerning the question asked by Tim Kohl,
about dealing with certain subspaces of group algebras,
I am not sure whether I understand what the goal is.

Perhaps we try a different approach, using the implementation
of cyclotomic fields instead of the algebraic extensions.
(Of course this is neither general enough nor practically desirable.)

Consider the following GAP session.

    gap> r2:= Sqrt( 2 );;                 # a square root of 2
    gap> K:= Field( Rationals, [ r2 ] );; # the field extension
    gap> S3:= SymmetricGroup( 3 );;
    gap> KS3:= GroupRing( K, S3 );;
    gap> emb:= Embedding( S3, KS3 );;
    gap> gens:= [ (1,2)^emb + (1,2,3)^emb, (1,3,2)^emb + (1,3)^emb ];
    [ (1)*(1,2)+(1)*(1,2,3), (1)*(1,3,2)+(1)*(1,3) ]

Now one wants to create a Q-space that is generated by some elements in KS3.
For that, we can either view the group algebra as a Q-space, ...

    gap> Q_KS3:= AsAlgebra( Rationals, KS3 );;
    gap> Dimension( Q_KS3 );
    12
    gap> Dimension( KS3 );
    6
    gap> H1:= Subspace( Q_KS3, gens );
    <vector space over Rationals, with 2 generators>

... or we form the vector space independent of the group algebra,
just by prescribing the base field and generators.
(The two variants are of course equal as sets.)

    gap> H2:= VectorSpace( Rationals, gens );
    <vector space over Rationals, with 2 generators>
    gap> H1 = H2;
    true

In both cases, forming products of elements in the subspaces works.

    gap> prod:= gens[1] * gens[2];
    (1)*()+(1)*(2,3)+(1)*(1,2)+(1)*(1,2,3)
    gap> prod in H;
    false

Is this roughly the setup of interest?
If yes then the analogous construction using general algebraic extensions
would require to deal with algebras/spaces over subfields of the extension.
Is that available?

All the best,
Thomas


On Thu, Oct 26, 2017 at 11:15:16AM -0400, tkohl at math.bu.edu wrote:
> 
> Dear forum,
> 
> This has gotten me part way to what I'm looking for.
> 
> (Many thanks Frank.)
> 
> But I'm running into a different problem now.
> 
> Basically, if one has, for example,
> 
> a:=Indeterminate(Rationals,"r");
> K:=AlgebraicExtension(Rationals,r^2-2)
> S3:=SymmetricGroup(3);
> KS3:=GroupRing(K, S3);
> emb:=Embedding(S3,KS3);
> 
> then I would like to be able to view KS3 as
> a module over Q, so that I can do something
> like this
> 
> H:=Subspace(KD3,[(1,2)^emb+(1,2,3)^emb, (1,3,2)^emb+(1,3)^emb ]);
> 
> so that H is the Q-span of { (1,2)+(1,2,3) , (1,3,2)+(1,3) }
> with the ultimate goal of being able to multiply elements
> of H and represent them with respect to this basis.
> 
> [I'm basically looking at Q-subalgebras of KG.]
> 
> Also, as an aside, the GaloisGroup() function seems
> not to be working. It gives the "no method found" error
> if I try to do GaloisGroup(K).  [I'm using 4r8.]
> 
> Thanks.
> 
> 	-T