# 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 : sag := sagbi transpose translation; i4 : netList first entries gens sag +------------------+ o4 = |w | | 3 | +------------------+ |w | | 2 | +------------------+ |w | | 1 | +------------------+ |t w + t w - v | | 2 1 3 2 3 | +------------------+ |t w + t w + v | | 1 2 2 3 1 | +------------------+ |t w - t w - v | | 1 1 3 3 2 | +------------------+ |w v + w v + w v | | 1 1 2 2 3 3| +------------------+

The above is precisely the 5 invariants Crook and Donelan 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 : SB = sagbi(sag1, Limit => 100); i9 : isSAGBI SB o9 = true i10 : netList first entries gens SB +---------------------------------------------------------------------------------------------------------------------+ | 2 2 2 | o10 = |x + x + x | | 7 8 9 | +---------------------------------------------------------------------------------------------------------------------+ |x x + x x + x x | | 4 7 5 8 6 9 | +---------------------------------------------------------------------------------------------------------------------+ | 2 2 2 | |x + x + x | | 4 5 6 | +---------------------------------------------------------------------------------------------------------------------+ |x x + x x + x x | | 1 7 2 8 3 9 | +---------------------------------------------------------------------------------------------------------------------+ |x x + x x + x x | | 1 4 2 5 3 6 | +---------------------------------------------------------------------------------------------------------------------+ | 2 2 2 | |x + x + x | | 1 2 3 | +---------------------------------------------------------------------------------------------------------------------+ |x x x - x x x - x x x + x x x + x x x - x x x | | 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 | |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| |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 | |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 | |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 | |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 | |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 | +---------------------------------------------------------------------------------------------------------------------+