[GAP Forum] working with GroupRings
Thomas Breuer
sam at Math.RWTH-Aachen.De
Thu Oct 26 17:04:07 BST 2017
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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