Dear Alper,
> On 04 Oct 2016, at 11:31, Alper Odabaş <aoda...@ogu.edu.tr> wrote:
>
> Dear forum,
>
>
>
> By linear algebra, the choice of an (ordered) basis for a free module of
> finite rank m yields an isomorphism to Z^{1 x m}, the module whose entries
> are row matrices with m columns.
>
>
>
> In GAP, how to get matrices from the group algebra
> GroupRing(GF(2),CyclicGroup(3))
Your question is a bit ambiguous, I'll interpret it as follows: Given a basis B
of an algebra A, how can I express an element x as a matrix over that basis?
Answer: Using the command AdjointMatrix. Here is an example:
gap> R:=GroupRing(GF(2),CyclicGroup(3));
<algebra-with-one over GF(2), with 1 generators>
gap> x:= R.1 + R.1^2;
gap> B:=Basis(R);
CanonicalBasis( <algebra-with-one over GF(2), with 1 generators> )
gap> AsList(B); # let's see which basis GAP picked...:
[ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1, (Z(2)^0)*f1^2 ]
gap> mat:=AdjointMatrix(Basis(R), x);
[ [ 0*Z(2), Z(2)^0, Z(2)^0 ], [ Z(2)^0, 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0,
0*Z(2) ] ]
gap> Display(mat);
. 1 1
1 . 1
1 1 .
Hope that helps,
Max
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