> restart;
> with(LinearAlgebra):
> with(Involutive):
> 
> pol:=proc(a,n)
> t^n+add((-1)^i*a[i]*t^(n-i),i=1..n);
> end proc:
> 
> REL:=proc(a,n,b,m,c)
> local A,B,C;
> A:=CompanionMatrix(pol(a,n),t);
> B:=CompanionMatrix(pol(b,m),t);
> C:=Matrix(convert(Matrix(n,n,(i,j)->A[i,j]*B),listlist));
> map(i->coeff(CharacteristicPolynomial(C,t)-pol(c,n*m),t,n*m-i),[$1..n*
> m]);
> end proc:
> 
> 
> 
# Relations
> L:=REL(a,2,b,3,c);

  L := [-a[1] b[1] + c[1],

                           2            2
        -2 a[2] b[2] + b[1]  a[2] + a[1]  b[2] - c[2],

                                                     3
        3 a[1] a[2] b[3] - b[1] a[2] a[1] b[2] - a[1]  b[3] + c[3],

               2                           2            2     2
        -2 a[2]  b[3] b[1] + a[2] b[1] a[1]  b[3] + a[2]  b[2]

                      2                            3     2
         - c[4], -a[2]  b[3] a[1] b[2] + c[5], a[2]  b[3]  - c[6]]

> L1:=subs([a[2]=1,b[3]=1],L)[1..-2];

                                          2       2
  L1 := [-a[1] b[1] + c[1], -2 b[2] + b[1]  + a[1]  b[2] - c[2],

                                      3
        3 a[1] - b[1] a[1] b[2] - a[1]  + c[3],

                           2       2
        -2 b[1] + b[1] a[1]  + b[2]  - c[4], -a[1] b[2] + c[5]]

> 
> 
> 
# Degrees of field extensions (determinant 1)
> indets(L1);

           {a[1], b[1], b[2], c[1], c[2], c[3], c[4], c[5]}

> vv:=[a[1], b[1], b[2]];

                       vv := [a[1], b[1], b[2]]

> with(combinat):
Warning, the protected name Chi has been redefined and unprotected

> c3a5:=choose(5,3);

  c3a5 := [[1, 2, 3], [1, 2, 4], [1, 2, 5], [1, 3, 4], [1, 3, 5],

        [1, 4, 5], [2, 3, 4], [2, 3, 5], [2, 4, 5], [3, 4, 5]]

> nops(c3a5);

                                  10

> CC:=proc(L)
> map(i->c[i],L);
> end proc:
> C3a5:=map(CC,c3a5);

  C3a5 := [[c[1], c[2], c[3]], [c[1], c[2], c[4]], [c[1], c[2], c[5]],

        [c[1], c[3], c[4]], [c[1], c[3], c[5]], [c[1], c[4], c[5]],

        [c[2], c[3], c[4]], [c[2], c[3], c[5]], [c[2], c[4], c[5]],

        [c[3], c[4], c[5]]]

> Comp:=proc(L)
> remove(has,map(i->c[i],[$1..5]),L);
> end proc:
> Comp(C3a5[1]);

                             [c[4], c[5]]

> Gra:=[]:
> for s from 1 to nops(C3a5) do  
> J:=InvolutiveBasisFast(subs(zip((i,j)->i=j,C3a5[s],[1,2,20]),map(k->L1
> [k],c3a5[s])),vv):
>   AssertInvBasis(J,vv):
>   print(C3a5[s],subs(t=1,PolHilbertSeries(t)));
>   Gra:=[op(Gra),subs(t=1,PolHilbertSeries(t))]:
> end do:

                        [c[1], c[2], c[3]], 7


                        [c[1], c[2], c[4]], 9


                        [c[1], c[2], c[5]], 3


                        [c[1], c[3], c[4]], 7


                        [c[1], c[3], c[5]], 4


                        [c[1], c[4], c[5]], 3


                        [c[2], c[3], c[4]], 18


                        [c[2], c[3], c[5]], 7


                        [c[2], c[4], c[5]], 9


                        [c[3], c[4], c[5]], 7

> Gra;

                   [7, 9, 3, 7, 4, 3, 18, 7, 9, 7]

# These are the degrees  [K(vv):K(C3a5[s])]. Since they are relatively
# prime, the c[i] generate K(vv) as a field.
> 
> 
> 
> 
# Degrees of field extensions (general determinant)
> L3 := subs(a[2]=1, L);

                                          2       2
  L3 := [-a[1] b[1] + c[1], -2 b[2] + b[1]  + a[1]  b[2] - c[2],

                                           3
        3 a[1] b[3] - b[1] a[1] b[2] - a[1]  b[3] + c[3],

                                2            2
        -2 b[3] b[1] + b[1] a[1]  b[3] + b[2]  - c[4],

                                    2
        -b[3] a[1] b[2] + c[5], b[3]  - c[6]]

> indets(L3);

     {a[1], b[1], b[2], b[3], c[1], c[2], c[3], c[4], c[5], c[6]}

> vv:=[a[1], b[1], b[2], b[3]];

                    vv := [a[1], b[1], b[2], b[3]]

> with(combinat):
> c4a6:=choose(6,4);

  c4a6 := [[1, 2, 3, 4], [1, 2, 3, 5], [1, 2, 3, 6], [1, 2, 4, 5],

        [1, 2, 4, 6], [1, 2, 5, 6], [1, 3, 4, 5], [1, 3, 4, 6],

        [1, 3, 5, 6], [1, 4, 5, 6], [2, 3, 4, 5], [2, 3, 4, 6],

        [2, 3, 5, 6], [2, 4, 5, 6], [3, 4, 5, 6]]

> CC:=proc(L)
> map(i->c[i],L);
> end proc:
> C4a6:=map(CC,c4a6);

  C4a6 := [[c[1], c[2], c[3], c[4]], [c[1], c[2], c[3], c[5]],

        [c[1], c[2], c[3], c[6]], [c[1], c[2], c[4], c[5]],

        [c[1], c[2], c[4], c[6]], [c[1], c[2], c[5], c[6]],

        [c[1], c[3], c[4], c[5]], [c[1], c[3], c[4], c[6]],

        [c[1], c[3], c[5], c[6]], [c[1], c[4], c[5], c[6]],

        [c[2], c[3], c[4], c[5]], [c[2], c[3], c[4], c[6]],

        [c[2], c[3], c[5], c[6]], [c[2], c[4], c[5], c[6]],

        [c[3], c[4], c[5], c[6]]]

> Comp:=proc(L)
> remove(has,map(i->c[i],[$1..6]),L);
> end proc:
> Comp(C4a6[1]);

                             [c[5], c[6]]

> Gra:=[]:
> for s from 1 to nops(C4a6) do  
> J:=InvolutiveBasisFast(subs(zip((i,j)->i=j,C4a6[s],[1,-3,7,20]),map(k-
> >L3[k],c4a6[s])),vv):
>   AssertInvBasis(J,vv):
>   print(C4a6[s],subs(t=1,PolHilbertSeries(t)));
>   Gra:=[op(Gra),subs(t=1,PolHilbertSeries(t))]:
> end do:

                     [c[1], c[2], c[3], c[4]], 10


                     [c[1], c[2], c[3], c[5]], 10


                     [c[1], c[2], c[3], c[6]], 14


                     [c[1], c[2], c[4], c[5]], 12


                     [c[1], c[2], c[4], c[6]], 18


                     [c[1], c[2], c[5], c[6]], 6


                     [c[1], c[3], c[4], c[5]], 10


                     [c[1], c[3], c[4], c[6]], 14


                     [c[1], c[3], c[5], c[6]], 8


                     [c[1], c[4], c[5], c[6]], 6


                     [c[2], c[3], c[4], c[5]], 24


                     [c[2], c[3], c[4], c[6]], 36


                     [c[2], c[3], c[5], c[6]], 14


                     [c[2], c[4], c[5], c[6]], 18


                     [c[3], c[4], c[5], c[6]], 14

> Gra;

      [10, 10, 14, 12, 18, 6, 10, 14, 8, 6, 24, 36, 14, 18, 14]

> 
> 
> 
> 
>