Some Magma Invariant Theory Calculations


{Magma code used: p = prime used; e.g. for the 3-dimensional rep.

K := GF(p);
sigma:=[1, 1, 0, 0, 1, 1, 0, 0, 1];
Cp := MatrixGroup<3,K|sigma>;
R:=InvariantRing(Cp);
PrimaryInvariants(R);
SecondaryInvariants(R);}

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
First, p = 3

2-dim'l representation:

primary:  x2, x1^3 + 2x_1x_2^2
secondary:  1

3-dim'l representation:

[
    x3,
    x1*x3 + x2^2 + 2*x2*x3,
    x1^3 + x1^2*x3 + 2*x1*x2^2 + x1*x2*x3
]
[
    1,
    x1*x3^2 + 2*x2^3 + x2^2*x3
]

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Now, p = 5

2-dim'l representation:

primary:  x2, x1^5 + 4*x_1*x_2^2
secondary:  1

3-dim'l representation:

[
    x3,
    x1*x3 + 2*x2^2 + 3*x2*x3,
    x1^5 + 2*x1^3*x3^2 + 2*x1^2*x2^2*x3 + 3*x1^2*x2*x3^2 + 4*x1*x2^4 +
        2*x1*x2^3*x3 + 4*x1*x2^2*x3^2 + 3*x2^4*x3 + 4*x2^3*x3^2 + 3*x2^2*x3^3
]
[
    1,
]

4-dim'l representation: 

[
    x4,
    x2*x4 + 2*x3^2 + 3*x3*x4,
    x1^2*x2*x4 + 2*x1^2*x3^2 + 3*x1^2*x3*x4 + 3*x1*x2^2*x3 + 2*x1*x2^2*x4 +
        2*x1*x2*x3^2 + 3*x1*x2*x3*x4 + x1*x3^3 + x1*x3^2*x4 + 3*x1*x3*x4^2 +
        3*x2^4 + 4*x2^3*x3 + x2^3*x4 + 3*x2*x3^3 + 4*x2*x3^2*x4 + 3*x2*x3*x4^2 +
        3*x3^4 + 4*x3^3*x4 + 3*x3^2*x4^2,
    x1^5 + x1^3*x2*x4 + 2*x1^3*x3^2 + 3*x1^3*x3*x4 + 2*x1^2*x2^2*x3 +
        3*x1^2*x2^2*x4 + 3*x1^2*x2*x3^2 + 2*x1^2*x2*x3*x4 + 4*x1^2*x3^3 +
        4*x1^2*x3^2*x4 + 2*x1^2*x3*x4^2 + 4*x1*x2^4 + 2*x1*x2^3*x3 +
        3*x1*x2^3*x4 + 4*x1*x2*x3^3 + 2*x1*x2*x3^2*x4 + 4*x1*x2*x3*x4^2 +
        4*x1*x3^4 + 2*x1*x3^3*x4 + 4*x1*x3^2*x4^2 + 4*x2^4*x3 + x2^4*x4 +
        2*x2^3*x3^2 + 3*x2^3*x3*x4 + 2*x2^2*x3^3 + x2^2*x3^2*x4 + 2*x2^2*x3*x4^2
        + 2*x2*x3^4 + 3*x2*x3^3*x4 + 4*x2*x3^2*x4^2 + x2*x3*x4^3 + 3*x3^3*x4^2 +
        3*x3^2*x4^3 + 4*x3*x4^4
]
[
    1,
    x1*x2*x4 + 2*x1*x3^2 + 3*x1*x3*x4 + 2*x1*x4^2 + 4*x2^2*x3 + x2^2*x4 +
        x2*x3^2 + 2*x2*x3*x4 + 2*x3^3 + 3*x3^2*x4,
    x1*x2*x4 + 2*x1*x3^2 + 3*x1*x3*x4 + 4*x2^2*x3 + x2^2*x4 + x2*x3^2 +
        4*x2*x3*x4 + 2*x2*x4^2 + 3*x3^3 + 2*x3^2*x4,
    x1^2*x4^2 + 3*x1*x2*x3*x4 + 2*x1*x2*x4^2 + 4*x1*x3^3 + 4*x1*x3^2*x4 +
        2*x1*x3*x4^2 + 2*x2^3*x4 + 3*x2^2*x3^2 + x2^2*x3*x4 + 2*x2^2*x4^2 +
        2*x2*x3^3 + 3*x2*x3*x4^2,
    x1^2*x2*x4^2 + 2*x1^2*x3^2*x4 + 3*x1^2*x3*x4^2 + 4*x1^2*x4^3 +
        3*x1*x2^2*x3*x4 + 2*x1*x2^2*x4^2 + 2*x1*x2*x3^2*x4 + 4*x1*x2*x4^3 +
        2*x1*x3^3*x4 + 4*x1*x3^2*x4^2 + 4*x1*x3*x4^3 + 2*x2^5 + 3*x2^4*x4 +
        4*x2^3*x3*x4 + 3*x2^3*x4^2 + x2^2*x3^2*x4 + 4*x2^2*x3*x4^2 + x2^2*x4^3 +
        3*x2*x3^4 + x2*x3^2*x4^2 + x2*x3*x4^3 + 4*x3^5 + 2*x3^4*x4 +
        4*x3^3*x4^2,
    x1^3*x4^2 + 2*x1^2*x2*x3*x4 + x1^2*x2*x4^2 + x1^2*x3^3 + 2*x1^2*x3^2*x4 +
        2*x1^2*x3*x4^2 + x1*x2^3*x4 + 4*x1*x2^2*x3^2 + 2*x1*x2^2*x3*x4 +
        4*x1*x2^2*x4^2 + x1*x2*x3^3 + 4*x1*x2*x3^2*x4 + 3*x1*x3^4 + 2*x1*x3^3*x4
        + x1*x3^2*x4^2 + 4*x1*x3*x4^3 + 4*x1*x4^4 + 4*x2^4*x3 + 2*x2^3*x3^2 +
        4*x2^3*x3*x4 + 4*x2^3*x4^2 + x2^2*x3^2*x4 + 4*x2^2*x3*x4^2 + 2*x2^2*x4^3
        + 4*x2*x3^4 + x2*x3^3*x4 + 4*x2*x3^2*x4^2 + 2*x2*x3*x4^3 + 4*x2*x4^4 +
        4*x3^5 + 2*x3^4*x4 + 4*x3^3*x4^2,
    x1^3*x2*x4^2 + 2*x1^3*x3^2*x4 + 3*x1^3*x3*x4^2 + 2*x1^2*x2^2*x3*x4 +
        x1^2*x2^2*x4^2 + 2*x1^2*x3^4 + x1^2*x3^2*x4^2 + 2*x1^2*x3*x4^3 +
        4*x1*x2^4*x4 + x1*x2^3*x3*x4 + 4*x1*x2^3*x4^2 + 3*x1*x2^2*x3^3 +
        2*x1*x2^2*x3*x4^2 + 2*x1*x2*x3^4 + 3*x1*x2*x3^3*x4 + 2*x1*x2*x3^2*x4^2 +
        3*x1*x2*x3*x4^3 + 3*x1*x2*x4^4 + x1*x3^5 + 4*x1*x3^4*x4 + 4*x1*x3^3*x4^2
        + 2*x1*x3^2*x4^3 + 4*x1*x3*x4^4 + x2^5*x3 + 3*x2^5*x4 + 3*x2^4*x3^2 +
        3*x2^4*x3*x4 + 4*x2^4*x4^2 + 4*x2^3*x3^3 + 4*x2^3*x3^2*x4 +
        2*x2^3*x3*x4^2 + 4*x2^3*x4^3 + x2^2*x3^3*x4 + 2*x2^2*x3^2*x4^2 +
        4*x2^2*x3*x4^3 + 3*x2^2*x4^4 + 2*x2*x3^5 + 3*x2*x3^3*x4^2 + x2*x3^2*x4^3
        + 4*x2*x3*x4^4,
    x1^4*x4^2 + x1^3*x2*x3*x4 + 2*x1^3*x2*x4^2 + 3*x1^3*x3^3 + 4*x1^3*x3^2*x4 +
        3*x1^3*x3*x4^2 + 2*x1^2*x2^3*x4 + 3*x1^2*x2^2*x3^2 + 2*x1^2*x2^2*x3*x4 +
        x1^2*x2^2*x4^2 + 2*x1^2*x2*x3^3 + 4*x1^2*x2*x3^2*x4 + 4*x1^2*x2*x3*x4^2
        + 3*x1^2*x2*x4^3 + 2*x1^2*x3^3*x4 + 3*x1^2*x3^2*x4^2 + 4*x1^2*x4^4 +
        x1*x2^4*x3 + x1*x2^4*x4 + 3*x1*x2^3*x3^2 + 2*x1*x2^3*x3*x4 +
        x1*x2^2*x3^3 + 4*x1*x2^2*x3^2*x4 + 4*x1*x2^2*x3*x4^2 + x1*x2^2*x4^3 +
        3*x1*x2*x3^3*x4 + 3*x1*x2*x3^2*x4^2 + x1*x2*x3*x4^3 + x1*x2*x4^4 +
        2*x1*x3^4*x4 + 4*x1*x3^3*x4^2 + x1*x3^2*x4^3 + 3*x1*x3*x4^4 + 4*x2^6 +
        x2^5*x3 + 4*x2^5*x4 + 2*x2^4*x3^2 + 3*x2^4*x3*x4 + 3*x2^4*x4^2 +
        x2^3*x3^3 + x2^3*x3^2*x4 + 2*x2^3*x4^3 + 4*x2^2*x3^3*x4 + 4*x2^2*x3*x4^3
        + 2*x2*x3^5 + 3*x2*x3^2*x4^3 + x3^5*x4 + 3*x3^4*x4^2 + x3^3*x4^3,
    x1^4*x2*x4^2 + 2*x1^4*x3^2*x4 + 3*x1^4*x3*x4^2 + x1^3*x2^2*x3*x4 +
        4*x1^3*x2^2*x4^2 + 4*x1^3*x2*x3^2*x4 + x1^3*x2*x3*x4^2 + 2*x1^3*x3^3*x4
        + 2*x1^3*x3^2*x4^2 + x1^3*x3*x4^3 + 3*x1^2*x2^4*x4 + 4*x1^2*x2^3*x3*x4 +
        x1^2*x2^3*x4^2 + 2*x1^2*x2^2*x4^3 + 3*x1^2*x2*x3^3*x4 +
        2*x1^2*x2*x3^2*x4^2 + x1^2*x2*x4^4 + x1^2*x3^4*x4 + 3*x1^2*x3^3*x4^2 +
        3*x1^2*x3^2*x4^3 + 3*x1^2*x3*x4^4 + 4*x1*x2^5*x3 + x1*x2^5*x4 +
        x1*x2^4*x3*x4 + 4*x1*x2^4*x4^2 + 3*x1*x2^3*x3^2*x4 + 3*x1*x2^3*x3*x4^2 +
        4*x1*x2^3*x4^3 + 4*x1*x2^2*x3^2*x4^2 + 4*x1*x2^2*x3*x4^3 +
        3*x1*x2^2*x4^4 + x1*x2*x3^5 + x1*x2*x3^4*x4 + 3*x1*x2*x3^3*x4^2 +
        2*x1*x2*x3^2*x4^3 + 3*x1*x2*x3*x4^4 + 4*x1*x3^5*x4 + 4*x1*x3^4*x4^2 +
        2*x1*x3^2*x4^4 + 3*x2^7 + 2*x2^6*x3 + x2^6*x4 + 2*x2^5*x3^2 +
        2*x2^5*x3*x4 + 3*x2^5*x4^2 + 2*x2^4*x3^2*x4 + x2^4*x3*x4^2 + 2*x2^3*x3^4
        + 3*x2^3*x3^3*x4 + 3*x2^3*x3^2*x4^2 + 4*x2^3*x4^4 + 3*x2^2*x3^5 +
        2*x2^2*x3^4*x4 + 4*x2^2*x3^3*x4^2 + 2*x2^2*x3^2*x4^3 + 4*x2^2*x3*x4^4 +
        3*x2*x3^6 + 2*x2*x3^5*x4 + 3*x2*x3^4*x4^2 + x2*x3^3*x4^3 + x2*x3^2*x4^4
        + 3*x3^7 + x3^6*x4 + 4*x3^5*x4^2 + 2*x3^4*x4^3,
    x1^4*x2*x4^3 + 2*x1^4*x3^2*x4^2 + 3*x1^4*x3*x4^3 + 3*x1^4*x4^4 +
        x1^3*x2^2*x3*x4^2 + 3*x1^3*x2*x3^2*x4^2 + 3*x1^3*x2*x4^4 +
        4*x1^3*x3^4*x4 + 3*x1^3*x3^3*x4^2 + 2*x1^3*x3^2*x4^3 + x1^3*x3*x4^4 +
        2*x1^3*x4^5 + 3*x1^2*x2^4*x4^2 + x1^2*x2^3*x3*x4^2 + 4*x1^2*x2^3*x4^3 +
        4*x1^2*x2^2*x3^3*x4 + 3*x1^2*x2^2*x3^2*x4^2 + 2*x1^2*x2^2*x3*x4^3 +
        x1^2*x2^2*x4^4 + 4*x1^2*x2*x3^4*x4 + 3*x1^2*x2*x3^2*x4^3 +
        2*x1^2*x2*x3*x4^4 + 3*x1^2*x2*x4^5 + 2*x1^2*x3^6 + 4*x1^2*x3^5*x4 +
        2*x1^2*x3^4*x4^2 + 3*x1^2*x3^3*x4^3 + 2*x1^2*x3^2*x4^4 + 2*x1^2*x3*x4^5
        + 4*x1*x2^6*x4 + 3*x1*x2^5*x3^2 + x1*x2^5*x3*x4 + 4*x1*x2^5*x4^2 +
        3*x1*x2^4*x3^2*x4 + x1*x2^4*x4^3 + 2*x1*x2^3*x3^3*x4 + x1*x2^3*x3^2*x4^2
        + 2*x1*x2^3*x3*x4^3 + 2*x1*x2^3*x4^4 + 3*x1*x2^2*x3^5 +
        3*x1*x2^2*x3^4*x4 + x1*x2^2*x3^3*x4^2 + 4*x1*x2^2*x3^2*x4^3 +
        4*x1*x2^2*x3*x4^4 + 4*x1*x2*x3^6 + x1*x2*x3^5*x4 + 2*x1*x2*x3^4*x4^2 +
        3*x1*x2*x3^3*x4^3 + 3*x1*x2*x4^6 + 3*x1*x3^7 + x1*x3^6*x4 + x1*x3^5*x4^2
        + 2*x1*x3^3*x4^4 + 2*x1*x3^2*x4^5 + x1*x3*x4^6 + 3*x1*x4^7 + x2^7*x3 +
        2*x2^7*x4 + 4*x2^6*x3^2 + 3*x2^6*x4^2 + 2*x2^5*x3^3 + x2^5*x3^2*x4 +
        x2^5*x3*x4^2 + 3*x2^4*x3^4 + x2^4*x3^3*x4 + 4*x2^4*x3^2*x4^2 +
        x2^4*x3*x4^3 + 4*x2^4*x4^4 + 3*x2^3*x3^5 + 4*x2^3*x3^4*x4 +
        4*x2^3*x3^3*x4^2 + x2^3*x3^2*x4^3 + 4*x2^3*x3*x4^4 + x2^2*x3^5*x4 +
        4*x2^2*x3^3*x4^3 + x2^2*x3^2*x4^4 + x2^2*x3*x4^5 + 3*x2^2*x4^6 +
        2*x2*x3^7 + x2*x3^5*x4^2 + 2*x2*x3^4*x4^3 + 2*x2*x3^3*x4^4 +
        3*x2*x3^2*x4^5 + 2*x2*x3*x4^6 + 3*x2*x4^7 + 3*x3^7*x4 + x3^6*x4^2 +
        3*x3^5*x4^3 + 4*x3^4*x4^4 + 4*x3^3*x4^5
]

5-dim'l representation:

Primary:  
[
    x5,
    x3*x5 + 2*x4^2 + 3*x4*x5,
    x1*x5 + 4*x2*x4 + 4*x2*x5 + 3*x3^2 + 2*x3*x4 + 4*x4^2 + x4*x5,
    x1^3*x5 + 2*x1^2*x2*x4 + 2*x1^2*x2*x5 + 4*x1^2*x3^2 + x1^2*x3*x4 +
        2*x1^2*x4^2 + 3*x1^2*x4*x5 + x1*x2^2*x3 + 4*x1*x2^2*x4 + 4*x1*x2*x3^2 +
        2*x1*x2*x3*x4 + x1*x2*x4^2 + 4*x1*x2*x4*x5 + 4*x1*x3^2*x4 + x1*x3^2*x5 +
        x1*x3*x4^2 + 4*x1*x3*x4*x5 + 3*x1*x4^3 + 3*x1*x4^2*x5 + 4*x1*x4*x5^2 +
        x2^4 + 3*x2^3*x3 + x2^3*x4 + 4*x2^3*x5 + x2^2*x3^2 + 2*x2^2*x4^2 +
        3*x2^2*x4*x5 + 4*x2*x3^2*x4 + 4*x2*x3^2*x5 + 3*x2*x3*x4^2 +
        2*x2*x3*x4*x5 + 2*x2*x4^3 + 3*x2*x4*x5^2 + 2*x3^3*x4 + x3^3*x5 +
        3*x3*x4^3 + x3*x4^2*x5 + x3*x4*x5^2 + 3*x4^4 + 4*x4^2*x5^2 + 3*x4*x5^3,
    x1^5 + x1^4*x5 + x1^3*x2*x4 + x1^3*x2*x5 + 2*x1^3*x3^2 + 3*x1^3*x3*x4 +
        x1^3*x4^2 + 4*x1^3*x4*x5 + 2*x1^2*x2^2*x3 + 3*x1^2*x2^2*x4 +
        3*x1^2*x2*x3^2 + 4*x1^2*x2*x3*x4 + 2*x1^2*x2*x4^2 + 3*x1^2*x2*x4*x5 +
        3*x1^2*x3^2*x4 + 2*x1^2*x3^2*x5 + 2*x1^2*x3*x4^2 + 3*x1^2*x3*x4*x5 +
        x1^2*x4^3 + x1^2*x4^2*x5 + 3*x1^2*x4*x5^2 + 4*x1*x2^4 + 2*x1*x2^3*x3 +
        4*x1*x2^3*x4 + x1*x2^3*x5 + 4*x1*x2^2*x3^2 + 3*x1*x2^2*x4^2 +
        2*x1*x2^2*x4*x5 + x1*x2*x3^2*x4 + x1*x2*x3^2*x5 + 2*x1*x2*x3*x4^2 +
        3*x1*x2*x3*x4*x5 + 3*x1*x2*x4^3 + 2*x1*x2*x4*x5^2 + 3*x1*x3^3*x4 +
        4*x1*x3^3*x5 + 2*x1*x3*x4^3 + 4*x1*x3*x4^2*x5 + 4*x1*x3*x4*x5^2 +
        2*x1*x4^4 + x1*x4^2*x5^2 + 2*x1*x4*x5^3 + 3*x2^4*x3 + 4*x2^4*x4 +
        4*x2^3*x3^2 + 4*x2^3*x3*x4 + 4*x2^3*x4^2 + x2^3*x4*x5 + 3*x2^2*x3^3 +
        3*x2^2*x3^2*x5 + x2^2*x4^3 + 4*x2^2*x4^2*x5 + 2*x2*x3^3*x4 +
        3*x2*x3^3*x5 + 2*x2*x3^2*x4^2 + 3*x2*x3^2*x4*x5 + 4*x2*x3*x4^3 +
        x2*x3*x4^2*x5 + 4*x2*x4^2*x5^2 + x2*x4*x5^3 + 4*x3^4*x4 + 4*x3^4*x5 +
        x3^3*x4^2 + 4*x3^3*x4*x5 + 2*x3^3*x5^2 + 4*x3^2*x4^3 + x3^2*x4^2*x5 +
        x3*x4^4 + 4*x3*x4^3*x5 + 4*x4^4*x5 + 2*x4^2*x5^3 + 4*x4*x5^4
]
Secondary output truncated by Magma calculator :(