# [GAP Forum] Convolutive algebra over a finite group

Stefano Simonucci stefano.simonucci at tin.it
Tue Aug 31 18:06:17 BST 2004

```On Mon, 2004-08-30 at 17:53, Thomas Breuer wrote:

> I am not sure whether Stefano means the group algebra,
> but this is closely related to the representations
> of a group is the group algebra.
>
> The GAP library contains basic support for creating group algebras,
> see section ``GroupRing'' in the GAP Reference Manual,
> for example `http://www.gap-system.org/Manuals/doc/htm/ref/CHAP063.htm'.
> Special functions for group rings can be found in the GAP package ``laguna'',
> see `http://www.gap-system.org/Packages/laguna.html'.
>
> For questions concerning the centre of a group algebra,
> also the character theory stuff in GAP may be of interest,
> see chapter ``Character Tables'' in the GAP Reference Manual,
> for example `http://www.gap-system.org/Manuals/doc/htm/ref/CHAP069.htm'.
>
> All the best,
> Thomas Breuer
>
>
I have resolved my problem by means of your suggestions.
For example

gap> A:=Group((1,2),(1,3));
Group([ (1,2), (1,3) ])
gap> B:=GroupRing(Rationals,A);
<algebra-with-one over Rationals, with 2 generators>
gap> dd:=DirectSumDecomposition(B);
[ <two-sided ideal in <algebra-with-one of dimension 6 over Rationals>,
(1 generators)>,
<two-sided ideal in <algebra-with-one of dimension 6 over Rationals>,
(1 generators)>,
<two-sided ideal in <algebra-with-one of dimension 6 over Rationals>,
(1 generators)> ]
gap> List(dd,Dimension);
[ 1, 1, 4 ]
gap> List(dd,Basis);
[ Basis( <two-sided ideal in <algebra-with-one of dimension 6 over
Rationals>, (dimension 1)>,
[ (1)*()+(-1)*(2,3)+(-1)*(1,2)+(1)*(1,2,3)+(1)*(1,3,2)+(-1)*(1,3) ]
),
Basis( <two-sided ideal in <algebra-with-one of dimension 6 over
Rationals>, (dimension 1)>,
[ (1)*()+(1)*(2,3)+(1)*(1,2)+(1)*(1,2,3)+(1)*(1,3,2)+(1)*(1,3) ] ),
Basis( <two-sided ideal in <algebra-with-one of dimension 6 over
Rationals>, (dimension 4)>,
[ (1)*()+(-1/2)*(1,2,3)+(-1/2)*(1,3,2),
(1)*(2,3)+(-2)*(1,2)+(1)*(1,3), (1)*(1,2)+(-1)*(1,3),
(1)*(1,2,3)+(-1)*(1,3,2) ] ) ]

Thank you
best, Stefano

```