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SubalgebraBases :: Example: Translation and rotation sub-actions of the adjoint action of SE(3)

Example: Translation and rotation sub-actions of the adjoint action of SE(3)

The following example shows how to use this package to calculate the invariants of the translation sub-action of the adjoint action of $SE(3)$, as considered in the preprint Polynomial invariants and SAGBI bases for multi-screws.

i1 : gndR = QQ[(t_1..t_3)|(w_1..w_3)|(v_1..v_3), MonomialOrder => Lex];
i2 : translation := matrix {{w_1}, {w_2}, {w_3}, {t_1*w_2+t_2*w_3+v_1}, {-t_1*w_1+t_3*w_3+v_2}, {-t_2*w_1-t_3*w_2+v_3}};

                6          1
o2 : Matrix gndR  <--- gndR
i3 : sag2 := sagbi transpose translation;
i4 : debugPrintMat gens sag2
   w
0:  3
   
   w
1:  2
   
   w
2:  1
   
   t w  + t w  - v
3:  2 1    3 2    3
   
   t w  + t w  + v
4:  1 2    2 3    1
   
   t w  - t w  - v
5:  1 1    3 3    2
   
   w v  + w v  + w v
6:  1 1    2 2    3 3
   

The above is precisely the 5 invariants Crook and Donelon give in equation (9), plus the additional 6th invariant.

The generators computed below verify Theorem 2 of Crook and Donelan, describing rotational invariants in the case where m=3.

i5 : R = QQ[x_1..x_9, MonomialOrder => Lex];
i6 : eqns := {x_1^2+x_2^2+x_3^2-1, x_1*x_4+x_2*x_5+x_3*x_6, x_1*x_7+x_2*x_8+x_3*x_9, x_1*x_4+x_2*x_5+x_3*x_6,
              x_4^2+x_5^2+x_6^2-1, x_4*x_7+x_5*x_8+x_6*x_9, x_1*x_7+x_2*x_8+x_3*x_9, x_4*x_7+x_5*x_8+x_6*x_9,
              x_7^2+x_8^2+x_9^2-1, x_1*x_5*x_9-x_1*x_6*x_8-x_2*x_4*x_9+x_2*x_6*x_7+x_3*x_4*x_8-x_3*x_5*x_7-1};
i7 : sag1 = subring sagbi eqns;
i8 : debugPrintMat gens sag1
    2    2    2
0: x  + x  + x
    7    8    9
   x x  + x x  + x x
1:  4 7    5 8    6 9
   
    2    2    2
2: x  + x  + x
    4    5    6
   x x  + x x  + x x
3:  1 7    2 8    3 9
   
   x x  + x x  + x x
4:  1 4    2 5    3 6
   
    2    2    2
5: x  + x  + x
    1    2    3
   x x x  - x x x  - x x x  + x x x  + x x x  - x x x
6:  1 5 9    1 6 8    2 4 9    2 6 7    3 4 8    3 5 7
   
    2 2    2 2                            2 2    2 2                2 2    2 2
7: x x  + x x  - 2x x x x  - 2x x x x  + x x  + x x  - 2x x x x  + x x  + x x
    4 8    4 9     4 5 7 8     4 6 7 9    5 7    5 9     5 6 8 9    6 7    6 8
        2        2                                         2        2                                         2        2
8: x x x  + x x x  - x x x x  - x x x x  - x x x x  + x x x  + x x x  - x x x x  - x x x x  - x x x x  + x x x  + x x x
    1 4 8    1 4 9    1 5 7 8    1 6 7 9    2 4 7 8    2 5 7    2 5 9    2 6 8 9    3 4 7 9    3 5 8 9    3 6 7    3 6 8
                            2        2        2                              2        2                   2
9: x x x x  + x x x x  - x x x  - x x x  - x x x  + x x x x  + x x x x  - x x x  - x x x  + x x x x  - x x x  + x x x x
    1 4 5 8    1 4 6 9    1 5 7    1 6 7    2 4 8    2 4 5 7    2 5 6 9    2 6 8    3 4 9    3 4 6 7    3 5 9    3 5 6 8
     2 2    2 2                            2 2    2 2                2 2    2 2
10: x x  + x x  - 2x x x x  - 2x x x x  + x x  + x x  - 2x x x x  + x x  + x x
     1 8    1 9     1 2 7 8     1 3 7 9    2 7    2 9     2 3 8 9    3 7    3 8
     2        2                                                    2        2                              2        2
11: x x x  + x x x  - x x x x  - x x x x  - x x x x  - x x x x  + x x x  + x x x  - x x x x  - x x x x  + x x x  + x x x
     1 5 8    1 6 9    1 2 4 8    1 2 5 7    1 3 4 9    1 3 6 7    2 4 7    2 6 9    2 3 5 9    2 3 6 8    3 4 7    3 5 8
     2 2    2 2                            2 2    2 2                2 2    2 2
12: x x  + x x  - 2x x x x  - 2x x x x  + x x  + x x  - 2x x x x  + x x  + x x
     1 5    1 6     1 2 4 5     1 3 4 6    2 4    2 6     2 3 5 6    3 4    3 5