On Sat, Nov 10, 2012 at 07:50:04PM +0800, Jianrong Li wrote:
> I use the following codes to list all elements in B3 Weyl group.
>
> Are there some other methods which can obtain the list of elements in Weyl
> group in GAP? Thank you very much.
>
> Best wishes,
> Jianrong.
>
> F:=FreeGroup("a", "b", "c");
> <free group on the generators [ a, b, c ]>
> gap> B:=F/[ F.1^2, F.2^2, F.3^2, (F.1*F.2)^4, (F.2*F.3)^3, (F.3*F.1)^2 ];
> <fp group on the generators [ a, b, c ]>
> gap> Size(B);
> 48
> gap> Elements(B);
> [ <identity ...>, a*b*a*b*c*b*a*b, c, c*b, a*b*a*b, b*a*b*c*b*a*b*c,
> a*b*a*b*c*b*a*b*c, a*b*a*c*b*a*b*c, b*a*b*c, a*c, b, a*c*b*a*b,
[...]
Dear Jianrong Li, dear Forum,
With GAPs current functionality for Weyl groups you could do the following.
# create the Weyl group in the default GAP:
W := WeylGroup( RootSystem( SimpleLieAlgebra("E", 6, Rationals) ) );
# after loading the 'QuaGroup' package it can be done more conveniently
LoadPackage("QuaGroup");
W := WeylGroup( RootSystem("E", 6) );
# The following function yields the elements of W as words which are
# represented as lists of generator indices.
# This function also shows how to generate the words one by one which
# may be interesting for large W.
WeylElements := function(W)
local rk, rho, o, res, wt;
rk := Length(GeneratorsOfGroup(W));
rho := List([1..rk], i-> 1);
o := WeylOrbitIterator(W, rho);
res := [];
while not IsDoneIterator(o) do
wt:=NextIterator(o);
Add(res, ConjugateDominantWeightWithWord(W,wt)[2]);
od;
return res;
end;
els := WeylElements(W);;
Length(els);
els[100];
# translating such a word into an element of W (a matrix):
wgens := GeneratorsOfGroup(W);;
mat := Product(wgens{els[100]}, One(W));
With best regards,
Frank Lübeck
--
/// Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64, ///
\\\ 52062 Aachen, Germany \\\
/// E-mail: frank.lueb...@math.rwth-aachen.de ///
\\\ WWW: http://www.math.rwth-aachen.de/~Frank.Luebeck/ \\\
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