[GAP Forum] Indeterminates and GroupRing elements

[Stefan Kohl]

Dear Forum,

Tim Kohl asked:

> I am trying to do the following with GroupRing()
>
> gap> QC3:=GroupRing(Rationals,Group((1,2,3)));;
> gap> x:=Indeterminate(Rationals,"x");;
> gap> y:=Indeterminate(Rationals,"y");;
> gap> z:=Indeterminate(Rationals,"z");;
> gap> B:=Basis(QC3);;
> gap> Elements(B);
> [ (1)*(), (1)*(1,2,3), (1)*(1,3,2) ]
>
> So far so good, but the following doesn't work.
>
> gap> h := x*B[1]+y*B[2]+y*B[3]
> Error, no method found! For debugging hints type ?Recovery from NoMethodFound
> Error, no 1st choice method found for `*' on 2 arguments called from
> <function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
>   called from read-eval loop at line 32 of *stdin*
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
>
> I want to do things like compute nilpotents etc by taking powers
> of 'generic' elements of a group ring (like h above) which will
> yield equations in the coefficients of the basis elements of the group ring.
>
> What is the correct way to do this?

I suppose what you would like to do is to compute in the
group ring Q[x,y,z]C_3, rather than in QC_3:

gap> R := PolynomialRing(Rationals,["x","y","z"]);
Rationals[x,y,z]
gap> RC3:=GroupRing(R,Group((1,2,3)));;
gap> B := Basis(RC3);;
gap> AsList(B);
[ (1)*(), (1)*(1,2,3), (1)*(1,3,2) ]
gap> h := x*B[1]+y*B[2]+y*B[3];
(x)*()+(y)*(1,2,3)+(y)*(1,3,2)

Does this help you?

Best regards,

     Stefan Kohl

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