restart
load "it.m2"

W =  QQ[x_1..x_6, a_1..a_6];
I = minors(2, matrix{{x_1, x_2, x_3}, {x_4, x_5, x_6}});
-- iterative
cc1 = {I=>1} | BMM(I,x_1,coDim=>codim I) 
-- there are 6 of these, mult(m)=0, where m=(x_1,...,x_6)

cc2 = BMM(cc1,x_2); -- 6, mult(m)=1
cc2' = BMM(cc1,x_4); -- 3, mult(m)=0
cc2'' = BMM(cc1,x_5); -- 6, mult(m)=1
cc3 = BMM(cc2,x_3); -- 2, mult(m)=2
cc3' = BMM(cc2,x_4); -- 12, mult(m)=1
cc3'' = BMM(cc2,x_6); -- 6, mult(m)=3
cc4 = BMM(cc3,x_4); -- 6, mult(m)=3
cc4' = BMM(cc3',x_5); -- 3, mult(m)=1
cc4'' = BMM(cc3',x_6); -- 6, mult(m)=3
cc5 = BMM(cc4,x_5); -- 6, mult(m)=3
cc6 = BMM(cc5,x_6); -- 1, mult(m)=3

scan(cc6,print) -- print the components

-- straightforward (pick f)
f = x_1;
f = x_1*x_2; -- (***) the components are correct, 
             -- but the multiplicities differ from the iterative computation
f = x_1*x_4;
f = x_1*x_5;
f = x_1*x_2*x_3;
f = x_1*x_2*x_4;
f = x_1*x_2*x_6;
f = x_1*x_2*x_3*x_4; -- takes too much time already
cc = {I=>1} | BMM(I,f,coDim=>codim I)