%off echo,nat$
off echo$
out "rreducparautommodg6_3case3N4.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of g_{6,3} as computed by calculautom6_3.red :"$

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

write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as in (Cohomology tables section 4.3.3) :
operator xi$
delta:=
mat((xi(1,1),xi(1,2),xi(1,3),0,0,0),
    (xi(2,1),xi(2,2),xi(2,3),0,0,0),
    (xi(3,1),xi(3,2),xi(3,3),0,0,0),
    (xi(4,1),xi(4,2),xi(4,3),xi(1,1)+xi(2,2),xi(2,3),-xi(1,3)),
    (xi(5,1),xi(5,2),xi(5,3),xi(3,2),xi(1,1)+xi(3,3),xi(1,2)),
    (xi(6,1),xi(6,2),xi(6,3),-xi(3,1),xi(2,1),xi(2,2)+xi(3,3)))$
write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({xi(1,1)=0,xi(1,2)=0,xi(1,3)=0,xi(2,1)=0,xi(2,2)=0,xi(2,3)=0,xi(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,1)=0,xi(4,2)=1,xi(4,3)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=1,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({xi(1,1)=0,xi(1,2)=0,xi(1,3)=0,xi(2,1)=0,xi(2,2)=0,xi(2,3)=0,xi(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,1)=-1,xi(4,2)=0,xi(4,3)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=1}, delta)$
matrix adx3(6,6)$
adx3:=
sub({xi(1,1)=0,xi(1,2)=0,xi(1,3)=0,xi(2,1)=0,xi(2,2)=0,xi(2,3)=0,xi(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,1)=0,xi(4,2)=0,xi(4,3)=0,xi(5,1)=-1,xi(5,2)=0,xi(5,3)=0,xi(6,1)=0,xi(6,2)=-1,xi(6,3)=0}, delta)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
write adx3:=adx3$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(4,1):=0$
xi(4,2):=0$
xi(5,1):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(4,1):=0,xi(4,2):=0,xi(5,1):=0."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "In the case 3, one has :"$
for i:=1:3 do for j:=1:3 do  xi(i,j):=0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(4,3),SS,xi(5,2),xi(5,3),SS,xi(6,1),xi(6,2),xi(6,3)}$
paramindexeslist:=
{ {4,3}, {5,2},{5,3}, {6,1},{6,2},{6,3}}$

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

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

%IF AUTOM=phi THEN  <<write "With the first kind automorphism one gets">>$
%IF AUTOM=psi THEN  <<write "With the second kind automorphism one gets">>$

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(AUTOM):=",det(AUTOM);
DELTAPRIMEMODADG:=M$
on nat$
write "DELTAPRIMEMODADG:=", M$
off nat$
CLEAR M$
%RETURN DELTAPRIMEMODADG$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************     SUBCASE 2 : we suppose xi(4,3) = 0   *********************"$
xi(4,3):=0$
write "xi(4,3):=",xi(4,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on factor$
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "We impose :"$
%b(1,1):=1$
b(1,2):=0$
write "b(1,2):=",b(1,2)$
b(1,3):=0$
write "b(1,3):=",b(1,3)$
%b(2,1):=0$
%b(2,2):=1$
b(2,3):=0$
write "b(2,3):=",b(2,3)$
%b(3,1):=1$
%b(3,2):=0$
%b(3,3):=1$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on factor$
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************     SUBCASE 2.2 : we suppose xi(5,3) = 0   *********************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(5,3):=0$
write "xi(5,3):=",xi(5,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************     SUBSUBCASE 2.2.2 : we suppose xi(5,2) = 0   ******************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(5,2):=0$
write "xi(5,2):=",xi(5,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on factor$
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "If moreover xi(6,3) neq 0, we may suppose xi(6,3):=1$ and we keep deltaprimemodg(6,3):=k by taking:"$
%deltaprimemodg(6,3):=b(2,2)*xi(6,3)$
b(2,2):=k/xi(6,3)$
write "b(2,2):=",b(2,2)$
write "and we get deltaprimemodg(6,2):=0 by taking:"$
%deltaprimemodg(6,2):= - b(3,2)*xi(6,3) + b(3,3)*xi(6,2)$
b(3,2):= b(3,3)*xi(6,2)/xi(6,3)$

write "b(3,2):=",b(3,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "and we get deltaprimemodg(6,1):=0 by taking:"$
%deltaprimemodg(6,1):=(k*( - b(3,1)*xi(6,3) + b(3,3)*xi(6,1)))/(b(1,1)*xi(6,3))$
b(3,1):=b(3,3)*xi(6,1)/xi(6,3)$
write "b(3,1):=",b(3,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence in that case we are reduced to :"$
write "shortformdeltaprimemodadg:={0,ss,0,0,ss,0,0,1}"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "On the other hand, if  xi(6,3) = 0, we get :"$
xi(6,3):=0$
write "xi(6,3):=",xi(6,3)$
clear   b(2,2),b(3,2),b(3,1)$
write "clear   b(2,2),b(3,2),b(3,1)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "In that latter case,  if  xi(6,2) NEQ 0, we may suppose xi(6,2)=1 :"$
xi(6,2):=1$
write "xi(6,2):=",xi(6,2)$
write "and we keep deltaprimemodg(6,2):=k by taking:"$
%deltaprimemodg(6,2):=b(3,3)*xi(6,2)$
b(3,3):=k/xi(6,2)$
write "b(3,3):=",b(3,3)$
write "and we get deltaprimemodg(6,1):=0 by taking:"$
%deltaprimemodg(6,1):=(( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(3,3))/b(1,1)$
b(2,1):= b(2,2)*xi(6,1)/xi(6,2)$
write "b(2,1):=",b(2,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence in that case we are reduced to :"$
write "shortformdeltaprimemodadg:={0,ss,0,0,ss,0,1,0}"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Finally,  suppose   xi(6,2) = 0:"$
xi(6,2):=0$
write "xi(6,2):=",xi(6,2)$
clear   b(2,1),b(3,3)$
write "clear   b(2,1),b(3,3)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "In that case we necessarily have xi(6,1) neq 0 since we do not conseder zero derivations"$
write "Hence in that case we are reduced to :"$
write "shortformdeltaprimemodadg:={0,ss,0,0,ss,1,0,0}"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
bye$