# 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