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
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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};
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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
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