[GAP Forum] spin group, finite subgroups

[Alexander Konovalov]

Dear Bartosz,

currently there is no immediate function in GAP to construct the  
Clifford
algebra. If I understood correctly what are you going to do, that  
maybe the
first step should be to find out how to do this.

I think you should be able to construct it using the  
AlgebraByStructureConstants
function, see http://www.gap-system.org/Manuals/doc/htm/ref/ 
CHAP060.htm#SSEC003.5

For example, this is how the QuaternionAlgebra function works, and  
quaternion
algebras could be regarded as a special case of Clifford algebras.  
You can have
a look on the function QuaternionAlgebra in gap4r4/lib/algsc.gi to  
get an idea
how to write your code.

When this will work, you should be able to generate subgroups by  
elements of
this algebra, for example:

gap> R:=QuaternionAlgebra(Rationals,1,1);
<algebra of dimension 4 over Rationals>
gap> gens:=GeneratorsOfAlgebra(R);
[ e, i, j, k ]
gap> G:=Group(gens);
#I  default `IsGeneratorsOfMagmaWithInverses' method returns `true'  
for [ e, i, j, k ]
<group with 4 generators>
gap> IdGroup(G);
[ 8, 3 ]

Also there is an undeposited implementation relevant to quaternion  
algebras
over the Rationals: http://www.geocities.com/assafwool/Quat/Quat.html.

Best wishes,
Alexander



On 14 Mar 2007, at 08:38, Bartosz Putrycz wrote:

> Dear Forum,
>
> I need to have some way to put/represent in GAP finite subgroups of  
> the
> spin group,
> given by generators in Clifford algebra,
> or given by their image in canonical epimorphism onto SO(n)
> For example:
> 1)
> <e_1 e_2, e_2 e_3, e_1 e_3> maped onto C2^2(diagonal +-1) in SO(3)
>
> 2)
> <1/sqrt{2} (1 - e_3 e_4), 1/sqrt{2} e_1(e_3 - e_4)> -> D8 in SO(4)
>
> Of course I could analyze these groups by myself, first is Q8,
> but I have more examples and I want to analyze them automatically  
> by GAP.
>
>
> Regards,
> Bartosz Putrycz.