%off echo,nat$
off echo$
out "rreducparautommodg6_2case2N1.r"$
operator b$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of g_{6,2} as computed by calculautom6_2.red :"$

phi:=
mat((b(1,1),0,0,0,0,0),(b(2,1),(b(4,4)*b(3,3) - b(4,3)*b(3,4))/b(1,1)**2,0,0,0,0
),(b(3,1),0,b(3,3),b(3,4),0,0),(b(4,1),0,b(4,3),b(4,4),0,0),(b(5,1),b(5,2),( - (
b(4,3)*b(3,1) - b(4,1)*b(3,3)))/b(1,1),( - (b(4,4)*b(3,1) - b(4,1)*b(3,4)))/b(1,
1),(b(4,4)*b(3,3) - b(4,3)*b(3,4))/b(1,1),0),(b(6,1),b(6,2),b(6,3),b(6,4),b(5,2)
*b(1,1),b(4,4)*b(3,3) - b(4,3)*b(3,4)))$

write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as in (Cohomology tables section 4.3.2) :
operator xi$
delta:=
mat((xi(1,1),0,0,0,0,0),
    (xi(2,1),xi(2,2),0,0,0,0),
    (xi(3,1),0,xi(3,3),xi(3,4),0,0),
    (xi(4,1),0,xi(4,3),2*xi(1,1)+xi(2,2)-xi(3,3),0,0),
    (xi(5,1),xi(5,2),xi(4,1),-xi(3,1),xi(1,1)+xi(2,2),0),
    (xi(6,1),xi(6,2),xi(6,3),xi(6,4),xi(5,2),2*xi(1,1)+xi(2,2)))$
write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,3)=0,xi(3,4)=0,xi(4,1)=0,xi(4,3)=0,xi(5,1)=0,xi(5,2)=1,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,4)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,3)=0,xi(3,4)=0,xi(4,1)=0,xi(4,3)=0,xi(5,1)=-1,xi(5,2)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,4)=0}, delta)$
matrix adx3(6,6)$
adx3:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,3)=0,xi(3,4)=0,xi(4,1)=0,xi(4,3)=0,xi(5,1)=0,xi(5,2)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,4)=1}, delta)$
matrix adx4(6,6)$
adx4:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,3)=0,xi(3,4)=0,xi(4,1)=0,xi(4,3)=0,xi(5,1)=0,xi(5,2)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=-1,xi(6,4)=0}, delta)$
matrix adx5(6,6)$
adx5:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,3)=0,xi(3,4)=0,xi(4,1)=0,xi(4,3)=0,xi(5,1)=0,xi(5,2)=0,xi(6,1)=-1,xi(6,2)=0,xi(6,3)=0,xi(6,4)=0}, delta)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
write adx3:=adx3$
write adx4:=adx4$
write adx5:=adx5$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic nilpotent derivation : the eigenvalues are 0"$
xi(1,1):=0$
xi(2,2):=0$
write "xi(1,1):=",xi(1,1)$
write "xi(2,2):=",xi(2,2)$
write "and in case 2 one has"$
xi(3,3):=0$
xi(3,4):=0$
xi(4,3):=0$
write "xi(3,3):=",xi(3,3)$
write "xi(3,4):=",xi(3,4)$
write "xi(4,3):=",xi(4,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(5,1):=0$
xi(5,2):=0$
xi(6,1):=0$
xi(6,3):=0$
xi(6,4):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(5,1):=0,xi(5,2):=0,xi(6,1):=0,xi(6,3):=0,xi(6,4):=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(2,1),SS,xi(3,1),SS,xi(4,1),SS,xi(6,2)}$
paramindexeslist:=
{ {2,1},{3,1},{4,1},{6,2}}$

write "shortformdelta:=", shortformdelta$
write "paramindexeslist:=",paramindexeslist$

off nat$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SHORTFORM(M0)$
BEGIN$
M:=M0$
WS:=  {M(2,1),SS,M(3,1),SS,M(4,1),SS,M(6,2)}$
RETURN WS$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE DELTAPRIMEMODADG(M0)$
BEGIN $
M:=M0$
M:=phi*M*phi**(-1)$
M:=M-M(5,2)*adx1 +M(5,1)*adx2 -M(6,4)*adx3 +M(6,3)*adx4 +M(6,1)*adx5$

write "shortformdeltaprimemodadg:=",shortform(M)$
for each U in paramindexeslist do <<write "deltaprimemodg(" ,part(U,1), ",", part(U,2), "):=" ,
                  M(part(U,1),part(U,2))>>$
write "det(phi):=",det(phi);
DELTAPRIMEMODADG:=M$
on nat$
write "DELTAPRIMEMODADG:=", M$
off nat$
CLEAR M$
%RETURN DELTAPRIMEMODADG$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%deltaprimemodg(3,1):=(b(3,4)*xi(4,1) + b(3,3)*xi(3,1))/b(1,1)$
%deltaprimemodg(4,1):=(b(4,4)*xi(4,1) + b(4,3)*xi(3,1))/b(1,1)$

write "************* SUBCASE 1: xi(3,1) and xi(4,1) not both zero. *********"$
write "Then one may suppose xi(3,1):=1 and xi(4,1):=0"$
xi(3,1):=1$
xi(4,1):=0$
write "xi(3,1):=",xi(3,1)$
write "xi(4,1):=",xi(4,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "To keep xi(3,1):=k,xi(4,1):=0, one has to take:"$
b(4,3):=0$
b(3,3):=b(1,1)*k$
write "b(4,3):=",b(4,3)$
write "b(3,3):=",b(3,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%deltaprimemodg(2,1):=(b(4,4)*xi(2,1)*k)/b(1,1)**2$
%deltaprimemodg(3,1):=k$
%deltaprimemodg(4,1):=0$
%deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in SUBCASE1 to:"$
write "shortformdeltaprime ={epsilon,SS,1,SS,0,SS,eta}"$
write "where epsilon =xi(2,1)=0,1"$
write "where eta =xi(6,2)=0,1"$
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************* SUBCASE 2: xi(3,1) = xi(4,1) =0. *********"$
xi(3,1):=0$
xi(4,1):=0$
write "xi(3,1):=",xi(3,1)$
write "xi(4,1):=",xi(4,1)$
clear b(4,3),b(3,3)$
write "clear b(4,3),b(3,3)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in SUBSUBCASE 2 to:"$
write "shortformdeltaprime ={epsilon,SS,0,SS,0,SS,1}"$
write "where epsilon =xi(2,1)=0,1"$
write "or shortformdeltaprime ={1,SS,0,SS,0,SS,0}"$
write "according to whether or not xi(6,2)=0"$
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%