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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