Dear GAP Forum,
Scott Moser asked
I have a rather simple question that i, unfortunately, could not find
in the manual. I am trying to compute the Molien series of finite
groups under different representations. Dr. Adler's post to the forum
was quite helpfull, but i'm affraid i do not know how to access specific
representaions of a group with it's CharacterTable.For example: gap> t:=CharacterTable(PSL(2,5)); CharacterTable( Group([ (3,5)(4,6), (1,2,5)(3,4,6) ]) ) gap> Display(t); CT12 2 2 . . . 3 1 . . . 1 5 1 . 1 1 .1a 2a 5a 5b 3a 2P 1a 1a 5b 5a 3a 3P 1a 2a 5b 5a 1a 5P 1a 2a 1a 1a 3aX.1 1 1 1 1 1 X.2 3 -1 A *A . X.3 3 -1 *A A . X.4 4 . -1 -1 1 X.5 5 1 . . -1
How would i go about obtaining the Molien series of the _second_
three-dimentional representation, labeled 'X.3'? In general, how do i
reference specific representaions presented in a CharacterTable?
Also, what about reducable representaions? Any idea how to direct sum
the Irreducable representations given in the tabe to form arbitrary
The Section ``MolienSeries'' in the GAP Reference Manual
should answer these questions.
In your example, a solution might look as follows.
gap> # Access the irreducible characters of the table. gap> irr:= Irr( t );; gap> # Compute the Molien series of the character 'X.3'. gap> MolienSeries( irr ); ( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 ) gap> # Compute the Molien series of a reducible character. gap> MolienSeries( irr + irr ); ( 1+z-z^3-z^4-z^5+z^7+z^8 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 )
I hope this helps.