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Fundamental Invariants


Fundamental 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 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.





Example RngInvar_FundamentalInvariants (H30E8)

We construct fundamental invariants for the invariant ring R of the
group G over Q, where G is permutation group consisting of two parallel
copies of S_3 in degree 6.  Notice that the sequence of fundamental
invariants is shorter and simpler than the sequence consisting of the
primary invariants combined with the secondary invariants.





> 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
]





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