%off echo,nat$
off echo$
out "rreducparautommodg6_20N1.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of g_{6,20} as computed by calculautom6_20.red :"$

phi:=
mat((b(1,1),0,0,0,0,0),(0,b(1,1)**2,0,0,0,0),(b(3,1),b(3,2),b(1,1)**3,0,0,0),(b(
4,1),b(3,2)**2/(2*b(1,1)**2),b(3,2)*b(1,1),b(1,1)**4,0,0),(b(5,1),b(5,2),(b(3,2)
**2 - 2*b(3,1)*b(1,1)**3)/(2*b(1,1)),b(3,2)*b(1,1)**2,b(1,1)**5,0),(b(6,1),b(6,2
),((2*b(4,1)*b(1,1)**2 - b(3,2)*b(3,1))*b(3,2) - 2*b(5,1)*b(1,1)**4)/(2*b(1,1)**
2),(b(4,1)*b(1,1)**2 - b(3,2)*b(3,1))*b(1,1), - b(3,1)*b(1,1)**4,b(1,1)**7))$



write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as in (Cohomology tables section 4.1.4) :
operator xi$
delta:=
mat((xi(1,1),0,0,0,0,0),
    (0,2*xi(1,1),0,0,0,0),
    (xi(3,1),xi(3,2),3*xi(1,1),0,0,0),
    (xi(4,1),0,xi(3,2),4*xi(1,1),0,0),
    (xi(5,1),xi(5,2),-xi(3,1),xi(3,2),5*xi(1,1),0),
    (xi(6,1),xi(6,2),-xi(5,1),xi(4,1),-xi(3,1),7*xi(1,1)))$
write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({xi(1,1)=0,xi(3,1)=0,xi(3,2)=1,xi(4,1)=0,xi(5,1)=0,xi(5,2)=0,xi(6,1)=0,xi(6,2)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({xi(1,1)=0,xi(3,1)=-1,xi(3,2)=0,xi(4,1)=0,xi(5,1)=0,xi(5,2)=0,xi(6,1)=0,xi(6,2)=0}, delta)$
matrix adx3(6,6)$
adx3:=
sub({xi(1,1)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=-1,xi(5,1)=0,xi(5,2)=-1,xi(6,1)=0,xi(6,2)=0}, delta)$
matrix adx4(6,6)$
adx4:=
sub({xi(1,1)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=0,xi(5,1)=-1,xi(5,2)=0,xi(6,1)=0,xi(6,2)=0}, delta)$
matrix adx5(6,6)$
adx5:=
sub({xi(1,1)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=0,xi(5,1)=0,xi(5,2)=0,xi(6,1)=0,xi(6,2)=-1}, delta)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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(1,1):=0$
write "xi(1,1):=",xi(1,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(3,1):=0$
xi(3,2):=0$
xi(4,1):=0$
xi(5,1):=0$
xi(6,2):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(5,1):=0,xi(6,2):=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(5,2),SS,xi(6,1)}$
paramindexeslist:=
{ {5,2},{6,1}}$

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

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

%IF AUTOM=phi THEN  <<write "With the first kind automorphism one gets">>$
IF AUTOM=phi THEN  <<write "With the generic 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);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****************** CASE 1 : xi(5,2) NEQ 0 *************************"$
write "Then one may suppose xi(5,2):=1 and one gets deltaprimemodg(4,2)=k by taking :"$
xi(5,2):=1$
write "xi(5,2):=",xi(5,2)$
%deltaprimemodg(5,2):=b(1,1)**3*xi(5,2)$
%deltaprimemodg(6,1):=b(1,1)**6*xi(6,1)$
b(1,1):=(k)**(1/3)$
write "b(1,1):=",b(1,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%deltaprimemodg(6,1):=xi(6,1)*k**2$
write "If xi(6,1) NEQ 0, we get deltaprime(6,1):=1 by taking :"$
k:=1/xi(6,1)$
write "k:=",k$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence, we are reduced in the case 1 under consideration to:"$
write "shortformdeltaprime ={1,SS,epsilon}"$
write "where epsilon=xi(6,1) =0,1."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****************** CASE 2 : xi(5,2) = 0 *************************"$
write "In that case, as we suppose delta NEQ 0 one necessarily has xi(6,1) NEQ 0"$
write "Hence we can suppose xi(6,1):=1,"$
write "and we are reduced  to:"$
write "shortformdeltaprime ={0,SS,1}."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%