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 fundamental invariants
for R is a generating set of R as an algebra over K.
FundamentalInvariants(R) : RngInvar -> [ RngMPolElt ]
Construct fundamental invariants for the invariant ring R=K[V]^G as a sorted sequence (with increasing degrees) of polynomials of R. The fundamental invariants are constructed by minimalizing the set of generators obtained by combining primary and secondary invariants for R.
> K := RationalField();
> G := PermutationGroup<6 | (1,2,3)(4,5,6), (1,2)(4,5)>;
> R := InvariantRing(G, K);
> PrimaryInvariants(R);
[
x1 + x2 + x3,
x4 + x5 + x6,
x1^2 + x2^2 + x3^2,
x4^2 + x5^2 + x6^2,
x1^3 + x2^3 + x3^3,
x4^3 + x5^3 + x6^3
]
> SecondaryInvariants(R);
[
1,
x1*x4 + x2*x5 + x3*x6,
x1^2*x4 + x2^2*x5 + x3^2*x6,
x1*x4^2 + x2*x5^2 + x3*x6^2,
x1^2*x4^2 + 2*x1*x2*x4*x5 + 2*x1*x3*x4*x6 + x2^2*x5^2 + 2*x2*x3*x5*x6 +
x3^2*x6^2,
x1^3*x4^3 + x1^2*x2*x4*x5^2 + x1^2*x3*x4*x6^2 + x1*x2^2*x4^2*x5 +
x1*x3^2*x4^2*x6 + x2^3*x5^3 + x2^2*x3*x5*x6^2 + x2*x3^2*x5^2*x6 +
x3^3*x6^3
]
> FundamentalInvariants(R);
[
1,
x1 + x2 + x3,
x4 + x5 + x6,
x1^2 + x2^2 + x3^2,
x1*x4 + x2*x5 + x3*x6,
x4^2 + x5^2 + x6^2,
x1^3 + x2^3 + x3^3 + x4*x5*x6,
x1^2*x4 + x2^2*x5 + x3^2*x6,
x1*x4^2 + x2*x5^2 + x3*x6^2,
x4^3 + x5^3 + x6^3
]