%off echo,nat$
off echo$
out "rreducparautommodg6_51xCN2.r"$
write "rreducparautommodg6_51xCN2.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of C x g_{5,1} as computed by calculautom6_51xC.red :"$
write "They fall into 4 kinds: "$
write "The first  kind (which contains the identity component) is: "$

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


write "The parameters are subject to the supplementary conditions :"$
condition11:=b(2,2)*( b(3,3)*b(1,1) - b(3,1)*b(1,3))$
write condition11,"neq 0"$


write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
write "condition11:=",condition11," neq 0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as computed by geneLplus.tex :
operator xi$

delta:=
mat((xi(1,1),xi(1,2),xi(1,3),xi(1,4),0,0),
(xi(2,1),xi(2,2),xi(1,4),xi(2,4),0,0),
(xi(3,1),xi(3,2),xi(3,3),-xi(2,1),0,0),
(xi(3,2),xi(4,2),-xi(1,2),xi(1,1)-xi(2,2)+xi(3,3),0,0),
(xi(5,1),xi(5,2),xi(5,3),xi(5,4),xi(1,1)+xi(3,3),xi(5,6)),
(xi(6,1),xi(6,2),xi(6,3),xi(6,4),0,xi(6,6)))$

write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({xi(1,1)=0,xi(1,2)=0,xi(1,3)=0,xi(1,4)=0,xi(2,1)=0,xi(2,2)=0,xi(2,4)=0,xi(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,2)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=1,xi(5,4)=0,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,4)=0,xi(6,6)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({xi(1,1)=0,xi(1,2)=0,xi(1,3)=0,xi(1,4)=0,xi(2,1)=0,xi(2,2)=0,xi(2,4)=0,xi(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,2)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=0,xi(5,4)=1,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,4)=0,xi(6,6)=0}, delta)$
matrix adx3(6,6)$
adx3:=
sub({xi(1,1)=0,xi(1,2)=0,xi(1,3)=0,xi(1,4)=0,xi(2,1)=0,xi(2,2)=0,xi(2,4)=0,xi(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,2)=0,xi(5,1)=-1,xi(5,2)=0,xi(5,3)=0,xi(5,4)=0,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,4)=0,xi(6,6)=0}, delta)$
matrix adx4(6,6)$
adx4:=
sub({xi(1,1)=0,xi(1,2)=0,xi(1,3)=0,xi(1,4)=0,xi(2,1)=0,xi(2,2)=0,xi(2,4)=0,xi(3,1)=0,xi(3,2)=0,xi(3,3)=0,xi(4,2)=0,xi(5,1)=0,xi(5,2)=-1,xi(5,3)=0,xi(5,4)=0,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=0,xi(6,4)=0,xi(6,6)=0}, delta)$
%matrix adx5(6,6)$
%adx5:=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
write adx3:=adx3$
write adx4:=adx4$
%write adx5:=adx5$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic nilpotent derivation : the eigenvalues are 0"$
xi(3,3):=-xi(1,1)$
write "xi(3,3):=",xi(3,3)$
xi(6,6):=0$
write "xi(6,6):=",xi(6,6)$

write "And the matrix A:="$
mat((xi(1,1),xi(1,2),xi(1,3),xi(1,4)),
(xi(2,1),xi(2,2),xi(1,4),xi(2,4)),
(xi(3,1),xi(3,2),xi(3,3),-xi(2,1)),
(xi(3,2),xi(4,2),-xi(1,2),xi(1,1)-xi(2,2)+xi(3,3)));
write "is nilpotent."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
M:=mat((xi(1,1),xi(1,2),xi(1,3),xi(1,4)),
(xi(2,1),xi(2,2),xi(1,4),xi(2,4)),
(xi(3,1),xi(3,2),xi(3,3),-xi(2,1)),
(xi(3,2),xi(4,2),-xi(1,2),xi(1,1)-xi(2,2)+xi(3,3)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "we may suppose that A is an element of sp(4,C)+, that is:"$
xi(1,1):=0$
xi(2,1):=0$
xi(2,2):=0$
xi(3,1):=0$
xi(3,2):=0$
xi(4,2):=0$
write "xi(1,1):=",xi(1,1)$
write "xi(2,1):=",xi(2,1)$
write "xi(2,2):=",xi(2,2)$
write "xi(3,1):=",xi(3,1)$
write "xi(3,2):=",xi(3,2)$
write "xi(4,2):=",xi(4,2)$


for j:=1:4 do <<write "M**",j,":=",M**j>>;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

off nat$
off factor$
for j:=1:4 do <<write "Trace(M**",j,")/2:=",Trace(M**j)/2>>;

write "************************************************************************"$
write "*******Suppose xi(1,2) neq 0."$
write "************************************************************************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear M;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(5,1):=0$
xi(5,2):=0$
xi(5,3):=0$
xi(5,4):=0$
write "By subtracting adjoints one then may suppose:"$
write "xi(5,1):=0,xi(5,2):=0,xi(5,3):=0,xi(5,4):=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(1,2),xi(1,3),xi(1,4),SS,xi(2,4),SS,xi(5,6),SS,xi(6,1),xi(6,2),xi(6,3),xi(6,4)}$
paramindexeslist:=
{{1,2},{1,3},{1,4},{2,4},{5,6},{6,1},{6,2},{6,3},{6,4}}$

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

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

IF AUTOM=phi THEN  <<write "With the first kind automorphism one gets">>$
IF AUTOM=psi THEN  <<write "With the second kind automorphism one gets">>$
IF AUTOM=rho THEN  <<write "With the third kind automorphism one gets">>$
IF AUTOM=chi THEN  <<write "With the fourth 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))>>$
on factor$
write "det(AUTOM):=",det(AUTOM);
DELTAPRIMEMODADG:=M$
%on nat$
off nat$
write "DELTAPRIMEMODADG:=", M$
off nat$
CLEAR M$
%RETURN DELTAPRIMEMODADG$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Take:"$
b(3,1):=0$
b(3,2):=0$
b(4,2):=0$
b(3,4):=0$
write "b(3,1):=",b(3,1)$
write "b(3,2):=",b(3,2)$
write "b(4,2):=",b(4,2)$
write "b(3,4):=",b(3,4)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then we get deltaprime(1,2)=1 , deltaprime(1,3)=0, deltaprimemodg(1,4)=0 and"$
write "deltaprime(6,2)=0 by taking:"$

%deltaprimemodg(1,2):=(b(1,1)*xi(1,2))/b(2,2)$
b(2,2):=(b(1,1)*xi(1,2))$
write "b(2,2):=",b(2,2)$

%deltaprimemodg(1,3):=(b(1,1)**2*xi(1,3) + 2*b(1,2)*b(1,1)*xi(1,4) + b(1,2)**2*xi
%(2,4) - 2*b(1,4)*b(1,1)*xi(1,2))/(b(3,3)*b(1,1))$

b(1,4):=(b(1,1)**2*xi(1,3) + 2*b(1,2)*b(1,1)*xi(1,4) + b(1,2)**2*xi
(2,4))/ ( 2*b(1,1)*xi(1,2))$
write "b(1,4):=",b(1,4)$

%deltaprimemodg(1,4):=(b(2,2)*b(1,1)*xi(1,4) + b(2,2)*b(1,2)*xi(2,4) - b(2,4)*b(1
%,1)*xi(1,2))/(b(3,3)*b(1,1))$
b(2,4):=(b(2,2)*b(1,1)*xi(1,4) + b(2,2)*b(1,2)*xi(2,4))/(b(1,1)*xi(1,2))$
write "b(2,4):=",b(2,4)$

%deltaprimemodg(6,2):=(b(6,1)*b(1,1)*xi(1,2) + b(6,6)*b(1,1)*xi(6,2) - b(6,6)*b(1
%,2)*xi(6,1))/(b(2,2)*b(1,1))$

b(6,1):=-( b(6,6)*b(1,1)*xi(6,2) - b(6,6)*b(1
,2)*xi(6,1))/(xi(1,2)*b(1,1))$
write "b(6,1):=",b(6,1)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then one gets deltaprimemodg(6,3)=0 by taking:"$
comment
deltaprimemodg(6,3):=(2*b(6,2)*b(1,1)**3*xi(1,4)*xi(1,2) + 2*b(6,2)*b(1,2)*b(1,1
)**2*xi(2,4)*xi(1,2) - 2*b(6,4)*b(1,1)**3*xi(1,2)**2 - 2*b(6,6)*b(1,1)**3*xi(6,2
)*xi(1,3) + 2*b(6,6)*b(1,1)**3*xi(6,3)*xi(1,2) + b(6,6)*b(1,2)*b(1,1)**2*xi(6,1)
*xi(1,3) - 2*b(6,6)*b(1,2)*b(1,1)**2*xi(6,2)*xi(1,4) + 2*b(6,6)*b(1,2)*b(1,1)**2
*xi(6,4)*xi(1,2) - b(6,6)*b(1,2)**3*xi(6,1)*xi(2,4) - 2*b(6,6)*b(1,3)*b(1,1)**2*
xi(6,1)*xi(1,2))/(2*b(3,3)*b(1,1)**3*xi(1,2))$

b(6,4):=(2*b(6,2)*b(1,1)**3*xi(1,4)*xi(1,2) + 2*b(6,2)*b(1,2)*b(1,1
)**2*xi(2,4)*xi(1,2)
%- 2*b(6,4)*b(1,1)**3*xi(1,2)**2
- 2*b(6,6)*b(1,1)**3*xi(6,2
)*xi(1,3) + 2*b(6,6)*b(1,1)**3*xi(6,3)*xi(1,2) + b(6,6)*b(1,2)*b(1,1)**2*xi(6,1)
*xi(1,3) - 2*b(6,6)*b(1,2)*b(1,1)**2*xi(6,2)*xi(1,4) + 2*b(6,6)*b(1,2)*b(1,1)**2
*xi(6,4)*xi(1,2) - b(6,6)*b(1,2)**3*xi(6,1)*xi(2,4) - 2*b(6,6)*b(1,3)*b(1,1)**2*
xi(6,1)*xi(1,2))/(2*b(1,1)**3*xi(1,2)**2)$
write "b(6,4):=",b(6,4)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "hence we may suppose :"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(1,2):=1$
xi(1,3):=0$
xi(1,4):=0$
write "xi(1,2):=",xi(1,2)$
write "xi(1,3):=",xi(1,3)$
write "xi(1,4):=",xi(1,4)$
xi(6,3):=0$
xi(6,2):=0$
write "xi(6,2):=",xi(6,2)$
write "xi(6,3):=",xi(6,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "**** Suppose xi(2,4) = 0."$
xi(2,4):=0$
write "xi(2,4):=",xi(2,4)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence we are reduced to shortformdelta:={1,0,0,ss,0,ss,1,ss,epsilon,0,0,eta}"$
write "where epsilon=xi(6,1)=0,1 and eta=xi(6,4)=0,1 in the case xi(5,6) neq 0."$
write "If xi(5,6) = 0, we are reduced to "$
write "shortformdelta:={1,0,0,ss,0,ss,0,ss,1,0,0,epsilon}"$
write "where epsilon=xi(6,4)=0,1 in the case xi(6,1) neq 0."$
write "or to shortformdelta:={1,0,0,ss,0,ss,0,ss,0,0,0,1}"$
write "in the case xi(6,1) = 0."$
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%