Dear GAP Forum,

Stefan Loewe wrote

(at least) the sources of GAP4 contain functions calculating Molien
Series. It would be very helpful for me if somebody could point me to
background material (theoretical or algorithmical) of the
implementation.

For the computation of the Molien series of a matrix representation
of a finite group G, only the eigenvalues of the matrices are needed,
and these can be computed from the character of the representation
together with information about the power maps.

So the question for a Molien series is interpreted as a question
to the character table of G.
This idea has been pointed out for example in

Joachim Neub"user, Herbert Pahlings, and Wilhelm Plesken,
CAS; design and use of a system for the handling of characters of
finite groups,
pp. 195--247 in
Michael~D. Atkinson, editor,
Computational Group Theory,
Proceedings LMS Symposium on Computational Group Theory,
Durham 1982, Academic Press, 1984.

This article lists three references, namely

J. M. Goethals and J. J. Seidel,
Spherical designs,
AMS Proc. Symp. Pure Math 34 (1979), pp. 255-272.

N. J. A. Sloane,
Error-correcting codes and invariant theory:
New applications of a nineteenth-century technique,
Amer. Math. Monthly 84 (1977), pp. 82-107.

Richard P. Stanley,
Invariants of finite groups and their applications to combinatorics,
Bull. Amer. Math. Soc. (New Ser.) 1 (1979), pp. 475-511.

The GAP implementation uses --besides the usual polynomials
and rational functions-- a special handling of rational functions
whose denominators are polynomials of the form (1-z^r)^k;
representing a Molien series as a sum of such rational functions
is useful for computing specific coefficients of the series
using a Taylor series for each summand.

This holds for both GAP 3 and GAP 4.
The only difference is that in GAP 4, rational functions are supported
and therefore a Molien series is regarded as a rational function.

Kind regards,
Thomas Breuer