%off echo,nat$
off echo$
out "rreducparautommodg6_n2case3.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of g_{6,n2} as computed by calculautom6_n2.red :"$


write "They fall into 2 kinds."$
write "First kind :"$
phi:=
mat((b(1,1),b(1,2),0,0,0,0),(b(2,1),b(2,2),0,0,0,0),(0,0,b(3,3),b(3,4),0,0),(0,0
,b(4,3),b(4,4),0,0),(b(5,1),b(5,2),b(5,3),b(5,4),b(2,2)*b(1,1) - b(2,1)*b(1,2),0
),(b(6,1),b(6,2),b(6,3),b(6,4),0,b(4,4)*b(3,3) - b(4,3)*b(3,4)))$
write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Second kind :"$
psi:=
mat((0,0,b(1,3),b(1,4),0,0),(0,0,b(2,3),b(2,4),0,0),(b(3,1),b(3,2),0,0,0,0),(b(4
,1),b(4,2),0,0,0,0),(b(5,1),b(5,2),b(5,3),b(5,4),0,b(2,4)*b(1,3) - b(2,3)*b(1,4)
),(b(6,1),b(6,2),b(6,3),b(6,4),b(4,2)*b(3,1) - b(4,1)*b(3,2),0))$

write "psi:=",psi;
on factor$
write "det(psi):=",det(psi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation  :
operator xi$
delta:=
mat((xi(1,1),xi(1,2),0,0,0,0),
    (xi(2,1),xi(2,2),0,0,0,0),
    (0,0,xi(3,3),xi(3,4),0,0),
    (0,0,xi(4,3),xi(4,4),0,0),
    (xi(5,1),xi(5,2),xi(5,3),xi(5,4),xi(1,1)+xi(2,2),0),
    (xi(6,1),xi(6,2),xi(6,3),xi(6,4),0,xi(3,3)+xi(4,4)))$
write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({xi(1,1)=0,xi(1,2)=0,xi(2,1)=0,xi(2,2)=0,xi(3,3)=0,xi(3,4)=0,xi(4,3)=0,xi(4,4)=0,xi(5,1)=0,xi(5,2)=1,xi(5,3)=0,xi(5,4)=0,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(1,2)=0,xi(2,1)=0,xi(2,2)=0,xi(3,3)=0,xi(3,4)=0,xi(4,3)=0,xi(4,4)=0,xi(5,1)=-1,xi(5,2)=0,xi(5,3)=0,xi(5,4)=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(1,2)=0,xi(2,1)=0,xi(2,2)=0,xi(3,3)=0,xi(3,4)=0,xi(4,3)=0,xi(4,4)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=0,xi(5,4)=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(1,2)=0,xi(2,1)=0,xi(2,2)=0,xi(3,3)=0,xi(3,4)=0,xi(4,3)=0,xi(4,4)=0,xi(5,1)=0,xi(5,2)=0,xi(5,3)=0,xi(5,4)=0,xi(6,1)=0,xi(6,2)=0,xi(6,3)=-1,xi(6,4)=0}, delta)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
write adx3:=adx3$
write adx4:=adx4$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic nilpotent derivation : "$
write "the matrix A:=((xi(1,1),xi(1,2)),xi(2,1),xi(2,2))) has to be nilpotent"$
write "the matrix B:=((xi(3,3),xi(3,4)),xi(4,3),xi(4,4))) has to be nilpotent"$
write "By Jordan reduction with a first kind automorphism, we get 4 cases"$
write "Case 1 : A:=0, B:=0"$
write "Case 2 : A:=((0,1),(0,0)), B:=0"$
write "Case 22 : A:=0, B:=((0,1),(0,0))"$
write "Case 3 : A:= B:=((0,1),(0,0))"$
write "The cases 2 and 22 are intertwined by some second kind automorphism"$
write "Hence we get only three case 1,2,3."$
write "We consider here the case 3."$
write "Then A:=((0,1),(0,0));  B:=((0,1),(0,0)) "$
xi(1,1):=0$
xi(1,2):=1$
xi(2,1):=0$
xi(2,2):=0$
for i:=1:2 do for j:=1:2 do <<write "xi(",i,",",j,"):=",xi(i,j)>>$
for i:=3:4 do for j:=3:4 do <<write "xi(",i,",",j,"):=",xi(i,j)>>$
xi(3,3):=0$
xi(3,4):=1$
xi(4,3):=0$
xi(4,4):=0$
write "delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(5,1):=0$
xi(5,2):=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(6,3):=0,xi(6,4):=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(5,3),xi(5,4),SS,xi(6,1),xi(6,2)}$
paramindexeslist:=
{ {5,3},{5,4},{6,1},{6,2}}$

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

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

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);
DELTAPRIMEMODADG(delta,psi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "To keep deltaprime(1,1):=0 and deltaprime(1,2):=1 under the first kind,"$
write "one has to take :"$
b(2,1):=0$
b(2,2):=b(1,1)$
write "b(2,2):=",b(2,2)$
write "b(2,1):=",b(2,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "To keep deltaprime(3,3):=0 and deltaprime(3,4):=1 under the first kind,"$
write "one has to take :"$
b(4,3):=0$
b(4,4):=b(3,3)$
write "b(4,3):=",b(4,3)$
write "b(4,4):=",b(3,3)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
DELTAPRIMEMODADG(delta,psi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "We get using the first kind deltaprime(5,4) and deltaprime(6,2):=0 if we take :"$
%deltaprimemodg(5,4):=( - ( - b(5,3)*b(3,3) + ( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3)
%)*b(1,1)**2))/b(3,3)**2$
b(5,3):= ( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3)
)*b(1,1)**2/b(3,3)$
write "b(5,3):=",b(5,3)$


%deltaprimemodg(6,2):=( - ( - b(6,1)*b(1,1) + ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1)
%)*b(3,3)**2))/b(1,1)**2$
b(6,1):= ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1)
)*b(3,3)**2/b(1,1)$
write "b(6,1):=",b(6,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence we may suppose xi(6,2):=0"$
xi(6,2):=0$
write "xi(6,2):=",xi(6,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then, we clearly are reduced in this case 3  to:"$
write "shortformdeltaprime ={eta,0,SS,epsilon,0}"$
write "where epsilon=xi(6,1)=0,1 . and eta =xi(5,3):=0,1"$
bye$