[GAP Forum] working with GroupRings

[tkohl at math.bu.edu]

Dear Thomas and forum.

This is indeed what I'm trying to do. 
Thanks for the code sample!

The Field() construction seems to behave
much nicer than AlgebraicExtension.

I'm going to push my luck a bit and ask
if I can adjoin 2^(1/3) since I want to work
in the splitting field of x^3-2.

As an aside, I found that back in 2001 or so
someone was trying to develop a Hopf algebra
package for GAP, but I couldn't find any further
mention of it. (I am looking at sub-algebras of
Hopf algebras.) 

	-T


On Thu, 26 Oct 2017, Thomas Breuer wrote:

> 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
> 
> 
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