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Primary Invariants


Primary Invariants






Let R=K[V]^G be the invariant ring of the group G over the field K and
suppose the degree of G is n.
A set of primary invariants of R is a set { f_1, ..., f_n }
of n algebraically independent homogeneous invariants of R
such that the invariant ring R is a
finitely generated module over A = K[f_1, ..., f_n].
A set of primary invariants always exists for any invariant ring R.
The invocation PrimaryInvariants(R) allows automatic construction of
primary invariants of R.  The primary invariants are stored in R and
recalled as necessary in subsequent computations.
The latest algorithm in Magma to compute primary invariants,
due to G. Kemper [G. Kemper, Calculating
Optimal Homogeneous Systems of Parameters, IWR Preprint 97-08,
Heidelberg 1997],
now guarantees that the degrees of the primary invariants found by the
algorithm are optimal (with respect to their product and then their sum).



PrimaryInvariants(R) : RngInvar -> [ RngMPolElt ]



Construct optimal primary invariants for the invariant ring R=K[V]^G as
a sorted sequence (with increasing degrees) of n polynomials of R where
n is the degree of G.





Example RngInvar_AdemMilgram (H30E6)

We compute primary invariants for the "first A_5 in SL(GF(2))" discussed
in A. Adem and R.J. Milgram: Cohomology of Finite Groups
(Berlin, Heidelberg, New York: Springer-Verlag).
The resulting degrees 3, 5, 8, and 12 are necessarily optimal
(cf. G. Kemper: Calculating Invariant Rings of Finite Groups over
Arbitrary Fields, cited at the beginning of the chapter).





> K := GF(2);
> G := MatrixGroup<4, K |
>     [0,1,0,0, 1,1,0,0, 0,0,1,1, 0,0,1,0],
>     [1,0,0,0, 0,1,0,0, 1,0,1,0, 0,1,0,1],
>     [1,0,1,0, 0,1,0,1, 0,0,1,0, 0,0,0,1]>;
> R := InvariantRing(G);
> time p := PrimaryInvariants(R);
Time: 1.399
> [TotalDegree(f): f in p];
[ 3, 5, 8, 12 ]





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