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);
CT1

2  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 3a

X.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
representations?

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[3] );
( 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[3] + irr[1] );
( 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.

Kind regards,
Thomas Breuer