[GAP Forum] working with GroupRings
tkohl at math.bu.edu
tkohl at math.bu.edu
Wed Oct 25 23:00:57 BST 2017
Dear Forum members,
This is somewhat related to a question I asked a while
back about GroupRings. My question is somewhat general,
but I will try to be as brief as possible.
I am trying to construct the group ring Q[w,z]S_3 where
Q[w,z] is the field extension of Q obtained by adjoining w,z where
w^3=2 and z is a primitive cube root of unity.
The method I am using is this:
z:=Indeterminate(Rationals,"z");
w:=Indeterminate(Rationals,"w");
R:=PolynomialRing(Rationals,["z","w"]);
I:=Ideal(R,[z^2+z+1,w^3-2]);
F:=R/I;
S3:=SymmetricGroup(3)
FS3:=GroupRing(F,S3);
so far so good.
One initial thing I notice is this:
gap> BF:=BasisVectors(Basis(F));
[ (1), (w), (w2), (z), (zw), (zw2) ]
which I can understand corresponds to the ideals 1+I, w+I, w^2+I, etc.
but I am not sure how to actually construct expressions by hand.
[That (zw2) is the representative instead of (z*w^2) is a bit jarring too, but that's
not the biggest issue.]
i.e. This does not work
gap> (w) in F;
false
although if I do
gap> BF[2] in F
then, of course, it is true.
Q1) How can I specify elements of F without having to refer to the literal list returned
from BasisVectors(Basis(F)) ?
Once I'm past this hurdle, I still want to work with elements of FS3 by taking linear
combinations of group elements and elements of F.
Q2) I want to be able to do something like this:
(z*(1,2))*(z*(1,2))
and have it give me (-z-1)*()
I know I need to use One(F) or One(FS3) in these expressions, but everything I have tried
ends up triggering
"Error, no method found! For debugging hints type ?Recovery from NoMethodFound"
Q3) Alternately, is there a way (like in Maple) to symbolically manipulate a polynomial
expression, for example
algsubs(z^2+z+1=0,z^4+z5)
and yield z+z^2?
(i.e. Forget about using a quotient ring and instead apply some regular expression
to 'mod out' by the relations w^3=2 and z^2+z+1=0.)
Pardon the length of my question, and thanks in advance for any assistance.
The main reason I'm using GAP in this instance is that Maple's grouptheory and
non-commuting variables infrastructure didn't work.
Thanks.
-Tim K.
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