[GAP Forum] List all elements in a Weyl group using GAP.

[Frank Lübeck]

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.Luebeck at Math.RWTH-Aachen.De                        ///
\\\  WWW:    http://www.math.rwth-aachen.de/~Frank.Luebeck/           \\\