%off echo,nat$
off echo$
out "rreducparautommodg6_11N3.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of g_{6,11} as computed by calculautom6_11.red :"$

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

write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as in (Cohomology tables section 4.2.5) :
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),xi(3,2),3*xi(1,1),0,0,0),
    (xi(4,1),xi(4,2),0,xi(1,1)+xi(2,2),0,0),
    (xi(5,1),xi(5,2),-xi(2,1),xi(4,2),2*xi(1,1)+xi(2,2),0),
    (xi(6,1),xi(6,2),xi(6,3),-xi(3,1)+xi(5,2),xi(4,2),3*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(3,2)=0,xi(4,1)=0,xi(4,2)=1,xi(5,1)=0,xi(5,2)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=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(3,2)=0,xi(4,1)=-1,xi(4,2)=0,xi(5,1)=0,xi(5,2)=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(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=0,xi(4,2)=0,xi(5,1)=0,xi(5,2)=0,xi(6,1)=0,xi(6,2)=-1,xi(6,3)=0}, delta)$
matrix adx4(6,6)$
adx4:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=0,xi(4,2)=0,xi(5,1)=-1,xi(5,2)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=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(3,2)=0,xi(4,1)=0,xi(4,2)=0,xi(5,1)=0,xi(5,2)=0,xi(6,1)=-1,xi(6,2)=0,xi(6,3)=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)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(4,1):=0$
xi(4,2):=0$
xi(5,1):=0$
xi(6,1):=0$
xi(6,2):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(6,1):=0,xi(6,2):=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),xi(3,2),SS,xi(5,2),SS,xi(6,3)}$
paramindexeslist:=
{ {2,1},{3,1},{3,2},{5,2},{6,3}}$

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

off nat$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SHORTFORM(M0)$
BEGIN$
M:=M0$
WS:=  {M(2,1),SS,M(3,1),M(3,2),SS,M(5,2),SS,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(6,2)*adx3 + M(5,1)*adx4 + M(6,1)*adx5$

%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$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************     SUBCASE 2 : we suppose xi(2,1) = 0   *********************"$
xi(2,1):=0$
write "xi(2,1):=",xi(2,1)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "********  SUBSUBCASE 2.1 : we suppose xi(3,2) NEQ 0   *********************"$
write "Then we can suppose  xi(3,2) :=1 ."$
xi(3,2):=1$
write "xi(3,2):=",xi(3,2)$
write "and we keep  deltaprime(3,2) :=k  by taking :"$
%deltaprimemodg(3,2):=(b(1,1)**3*xi(3,2))/b(2,2)$
b(2,2):=(b(1,1)**3*xi(3,2))/k$
write "b(2,2):=",b(2,2)$
write "One then  gets deltaprime(3,1)=0  by taking:"$
%deltaprimemodg(3,1):=(( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1)**2)/b(2,2)$
b(2,1):= b(2,2)*xi(3,1)/xi(3,2)$
write "b(2,1):=",b(2,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Now, if xi(6,3) NEQ 0 we get deltaprime(6,3)=k by taking:"$
%deltaprimemodg(6,3):=(b(1,1)**3*xi(6,3))/k$
b(1,1):=(k**2/xi(6,3))**(1/3)$
write "b(1,1):=",b(1,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in SUBSUBCASE 2.1   to:"$
write "shortformdeltaprime ={0,SS,0,1,SS,epsilon,SS,eta}"$
write "where epsilon =xi(5,2)=0,1 and eta=xi(6,3)=0,1."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "********  SUBSUBCASE 2.2 : we suppose xi(3,2) = 0   *********************"$
xi(3,2):=0$
write "xi(3,2):=",xi(3,2)$
clear b(2,2),b(2,1),b(1,1)$
write "clear b(2,2),b(2,1),b(1,1)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Now, if xi(3,1) NEQ 0 can suppose xi(3,1):=1"$
xi(3,1):=1$
write "xi(3,1):=",xi(3,1)$
write "and we keep deltaprime(3,1)=k by taking:"$
%deltaprimemodg(3,1):=b(1,1)**2*xi(3,1)$
b(1,1):=sqrt(k/xi(3,1))$
write "b(1,1):=",b(1,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in that case to:"$
write "shortformdeltaprime ={0,SS,1,0,SS,a,SS,epsilon}"$
write "where a =xi(5,2) are any complex number and epsilon=xi(6,3)=0,1"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Suppose now  xi(3,1) = 0."$
xi(3,1):=0$
write "xi(3,1):=",xi(3,1)$
clear b(1,1)$
write "clear b(1,1)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then, we are reduced according to whether or not  xi(5,2) NEQ  0  to:"$
write "shortformdeltaprime ={0,SS,0,0,SS,1,SS,epsilon}"$
write "where epsilon =xi(6,3)=0,1"$
write "or:"$
write "shortformdeltaprime ={0,SS,0,0,SS,0,SS,1}"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%