My understanding is that GAP code has been written
by (or under the supervision of) Prof. Pahlings at
Aachen - for the generalized Molien series with
denominator Chi[i](g^-1). Perhaps someone can shed
more light on this?
A good test example is the group of isometries of the
Leech lattice (dim = 24, |G| = 8315553613086720000).
Careful coding is needed for large groups.
[For Molien series,see Slo7 in SLAG = Amer Math Monthly 84 (1977) pp
82-107.]
John McKay
On Sat, 12 May 2007, David Joyner wrote:
> Hi:
>
> With kind help from Laurent Bartholdi, here is the answer to
> a question I recently sent to the GAP Support about Molien series.
> Specifically, how is GAP's command MolienSeries for a permutation group
> related to the function
>
> M(x) = (1/|G|)\sum_{g\in G} det(1-x*g)^(-1)
>
> which some call the Molien series of G.
>
> This is answered in the case of permutation groups having a
> transitive action on a set X.
>
> A group G, given as a permutation group on n points, has a "natural"
> representation of dimension n, given by permutation matrices. The
> "usual" Molien series is the one associated to that representation.
> Character values then count fixed points of the corresponding
> permutations.
>
> You have to realize the function M(x) as the MolienSeries
> attached to the "natural" (or permutation) character of G.
> Here is an example of the syntax used to compute M(x) using GAP:
>
>
> gap> pi := NaturalCharacter( SymmetricGroup(3) );
> Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 3, 1, 0 ] )
> gap> ConstituentsOfCharacter( pi );
> [ Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] ),
> Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, -1 ] ) ]
> gap> irr:= Irr( SymmetricGroup( 3 ) );
> [ Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, -1, 1 ] ),
> Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, -1 ] ),
> Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] ) ]
> gap> MolienSeries( irr[2]+irr[3] );
> ( 1 ) / ( (1-z^3)*(1-z^2)*(1-z) )
>
>
> Indeed, a direct computation verifies that
> M(x) = 1 / ( (1-x^3)*(1-x^2)*(1-x) )
>
> For groups other than the symmetric group, you may need to replace
> NaturalCharacter by PermutationCharacter.
>
> - David Joyner and Laurent Bartholdi
>
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