I would like to announce the GAP3 file PrimitiveInvariant.g, which
contains programs to compute primitive invariants, polynomials that
characterize subgroups of the symmetric group and that are of use in
Galois theory. The main algorithms encoded by the programs generalize
methods of K. Girstmair for calculating minimal primitive invariants.

Let L and H be subgroups of the symmetric group of degree n, with H
contained in L. The function MinimalPrimitiveInvariants(n,L,H) computes
all representatives of H-invariant L-primitive polynomials of minimal
degree. The function AllPrimitiveInvariants(n,L,H) computes all
representatives of H-invariant L-primitive polynomials of degree up to
n(n-1)/2. In applications to resolvant calculation in Galois theory, n is
the degree of the polynomial, L the candidate group and H the test group.

The file is divided into two parts. The first part contains functions for
calculating with combinations of lists that represent indices of monomials
in n variables. These lists give a GAP3 way to handle polynomials, which
are dealt with differently in GAP4. The second part of the file contains
functions to compute what one might call the essential partitions and
sets, as well as the main functions.

PrimitiveInvariant.g is available in the "deposit" ftp subdirectory, as
pub/gap/gap-3.4.4/deposit/gap/priminv.g
or by following the links "Get GAP...more" and "GAP3 User contributions"
from the GAP main Web page, to arrive at the URL
www-history.mcs.st-and.ac.uk/~gap/Info/deposit.html.
The article "Calculs d'Invariants Primitifs de Groupes finis" which gives
the theory behind the algorithms will appear in RAIRO -Informatique
Theorique et Programmation and is also available at
www-calfor.lip6.fr/~abdeljao.

I hope that GAP users will find these programs useful. I would be glad to
correspond further with anybody who has questions about them or has
applications for them.

Ines Abdeljaouad

< Ines.Abdeljaouad@lip6.fr >