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), 3 (b(3,1),b(3,2),b(1,1) ,0,0,0), (b(4,1),b(4,2),0,b(2,2)*b(1,1),0,0), 2 2 (b(5,1),b(5,2), - b(2,1)*b(1,1) ,b(4,2)*b(1,1),b(2,2)*b(1,1) ,0), (b(6,1),b(6,2),b(6,3), - b(3,1)*b(2,2) + b(3,2)*b(2,1) + b(5,2)*b(1,1), 2 3 b(4,2)*b(1,1) ,b(2,2)*b(1,1) )) 4 10 det(phi):=b(2,2) *b(1,1) generic derivation : 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))) [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx1 := [ ] [0 1 0 0 0 0] [ ] [0 0 0 1 0 0] [ ] [0 0 0 0 1 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx2 := [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 1 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx3 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 -1 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx5 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 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 phi:= mat((b(1,1),0,0,0,0,0), (b(2,1),b(2,2),0,0,0,0), 3 (b(3,1),b(3,2),b(1,1) ,0,0,0), (b(4,1),b(4,2),0,b(2,2)*b(1,1),0,0), 2 2 (b(5,1),b(5,2), - b(2,1)*b(1,1) ,b(4,2)*b(1,1),b(2,2)*b(1,1) ,0), (b(6,1),b(6,2),b(6,3),b(5,2)*b(1,1) - b(3,1)*b(2,2) + b(3,2)*b(2,1), 2 3 b(4,2)*b(1,1) ,b(2,2)*b(1,1) )) 4 10 det(phi):=b(2,2) *b(1,1) delta:= [ 0 0 0 0 0 0] [ ] [xi(2,1) 0 0 0 0 0] [ ] [xi(3,1) xi(3,2) 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 xi(5,2) - xi(2,1) 0 0 0] [ ] [ 0 0 xi(6,3) - xi(3,1) + xi(5,2) 0 0] 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}} shortformdeltaprimemodadg:={(b(2,2)*xi(2,1))/b(1,1), ss, (b(3,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1)**3)/(b(2,2) *b(1,1)), (b(1,1)**3*xi(3,2))/b(2,2), ss, ( - b(2,1)*b(1,1)**3*xi(3,2) + (b(1,1)**3*xi(5,2) + b(3,2)*xi(2,1))*b(2,2))/(b(2 ,2)*b(1,1)), ss, b(2,2)*xi(6,3)}$ deltaprimemodg(2,1):=(b(2,2)*xi(2,1))/b(1,1)$ deltaprimemodg(3,1):=(b(3,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1 ))*b(1,1)**3)/(b(2,2)*b(1,1))$ deltaprimemodg(3,2):=(b(1,1)**3*xi(3,2))/b(2,2)$ deltaprimemodg(5,2):=( - b(2,1)*b(1,1)**3*xi(3,2) + (b(1,1)**3*xi(5,2) + b(3,2)* xi(2,1))*b(2,2))/(b(2,2)*b(1,1))$ deltaprimemodg(6,3):=b(2,2)*xi(6,3)$ det(AUTOM):=b(2,2)**4*b(1,1)**10$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), b(2,2)*xi(2,1) (----------------,0,0,0,0,0), b(1,1) 3 b(3,2)*b(2,2)*xi(2,1) + ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1) (----------------------------------------------------------------------, b(2,2)*b(1,1) 3 b(1,1) *xi(3,2) -----------------,0,0,0,0), b(2,2) (0,0,0,0,0,0), 3 3 - b(2,1)*b(1,1) *xi(3,2) + (b(1,1) *xi(5,2) + b(3,2)*xi(2,1))*b(2,2) (0,-----------------------------------------------------------------------, b(2,2)*b(1,1) - b(2,2)*xi(2,1) -------------------,0,0,0), b(1,1) 2 (0,0,b(2,2)*xi(6,3),( - xi(3,1) + xi(5,2))*b(1,1) ,0,0)) ************ SUBCASE 2 : we suppose xi(2,1) = 0 *********************$ xi(2,1):=0$ shortformdeltaprimemodadg:={0, ss, (( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1)**2)/b(2,2), (b(1,1)**3*xi(3,2))/b(2,2), ss, (( - b(2,1)*xi(3,2) + b(2,2)*xi(5,2))*b(1,1)**2)/b(2,2), ss, b(2,2)*xi(6,3)}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=(( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1)**2)/b(2,2)$ deltaprimemodg(3,2):=(b(1,1)**3*xi(3,2))/b(2,2)$ deltaprimemodg(5,2):=(( - b(2,1)*xi(3,2) + b(2,2)*xi(5,2))*b(1,1)**2)/b(2,2)$ deltaprimemodg(6,3):=b(2,2)*xi(6,3)$ det(AUTOM):=b(2,2)**4*b(1,1)**10$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), 2 3 ( - b(2,1)*xi(3,2) + b(2,2)*xi(3,1))*b(1,1) b(1,1) *xi(3,2) (----------------------------------------------,-----------------,0,0,0,0), b(2,2) b(2,2) (0,0,0,0,0,0), 2 ( - b(2,1)*xi(3,2) + b(2,2)*xi(5,2))*b(1,1) (0,----------------------------------------------,0,0,0,0), b(2,2) 2 (0,0,b(2,2)*xi(6,3),( - xi(3,1) + xi(5,2))*b(1,1) ,0,0)) ******** SUBSUBCASE 2.1 : we suppose xi(3,2) NEQ 0 *********************$ Then we can suppose xi(3,2) :=1 .$ xi(3,2):=1$ and we keep deltaprime(3,2) :=k by taking :$ b(2,2):=b(1,1)**3/k$ One then gets deltaprime(3,1)=0 by taking:$ b(2,1):=(b(1,1)**3*xi(3,1))/k$ shortformdeltaprimemodadg:={0, ss, 0, k, ss, ( - xi(3,1) + xi(5,2))*b(1,1)**2, ss, (b(1,1)**3*xi(6,3))/k}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=k$ deltaprimemodg(5,2):=( - xi(3,1) + xi(5,2))*b(1,1)**2$ deltaprimemodg(6,3):=(b(1,1)**3*xi(6,3))/k$ det(AUTOM):=b(1,1)**22/k**4$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), (0,k,0,0,0,0), (0,0,0,0,0,0), 2 (0,( - xi(3,1) + xi(5,2))*b(1,1) ,0,0,0,0), 3 b(1,1) *xi(6,3) 2 (0,0,-----------------,( - xi(3,1) + xi(5,2))*b(1,1) ,0,0)) k Now, if xi(6,3) NEQ 0 we get deltaprime(6,3)=k by taking:$ b(1,1):=(k**2/xi(6,3))**(1/3)$ shortformdeltaprimemodadg:={0, ss, 0, k, ss, (k**2/xi(6,3))**(2/3)*( - xi(3,1) + xi(5,2)), ss, k}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=k$ deltaprimemodg(5,2):=(k**2/xi(6,3))**(2/3)*( - xi(3,1) + xi(5,2))$ deltaprimemodg(6,3):=k$ det(AUTOM):=((k**2/xi(6,3))**(1/3)*k**10)/xi(6,3)**7$ DELTAPRIMEMODADG:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 k 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [ 2 ] [ k 2/3 ] [0 (---------) *( - xi(3,1) + xi(5,2)) 0 0 0 0] [ xi(6,3) ] [ ] [ 2 ] [ ( - xi(3,1) + xi(5,2))*k ] [0 0 k --------------------------- 0 0] [ 2 ] [ k 1/3 ] [ (---------) *xi(6,3) ] [ xi(6,3) ] Hence, we are reduced in SUBSUBCASE 2.1 to:$ shortformdeltaprime ={0,SS,0,1,SS,epsilon,SS,eta}$ where epsilon =xi(5,2)=0,1 and eta=xi(6,3)=0,1.$ ******** SUBSUBCASE 2.2 : we suppose xi(3,2) = 0 *********************$ xi(3,2):=0$ clear b(2,2),b(2,1),b(1,1)$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(3,1), 0, ss, b(1,1)**2*xi(5,2), ss, b(2,2)*xi(6,3)}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=b(1,1)**2*xi(3,1)$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,2):=b(1,1)**2*xi(5,2)$ deltaprimemodg(6,3):=b(2,2)*xi(6,3)$ det(AUTOM):=b(2,2)**4*b(1,1)**10$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), 2 (b(1,1) *xi(3,1),0,0,0,0,0), (0,0,0,0,0,0), 2 (0,b(1,1) *xi(5,2),0,0,0,0), 2 (0,0,b(2,2)*xi(6,3),( - xi(3,1) + xi(5,2))*b(1,1) ,0,0)) Now, if xi(3,1) NEQ 0 can suppose xi(3,1):=1$ xi(3,1):=1$ and we keep deltaprime(3,1)=k by taking:$ b(1,1):=sqrt(k)$ shortformdeltaprimemodadg:={0, ss, k, 0, ss, xi(5,2)*k, ss, b(2,2)*xi(6,3)}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=k$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,2):=xi(5,2)*k$ deltaprimemodg(6,3):=b(2,2)*xi(6,3)$ det(AUTOM):=b(2,2)**4*k**5$ DELTAPRIMEMODADG:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [k 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 xi(5,2)*k 0 0 0 0] [ ] [0 0 b(2,2)*xi(6,3) ( - 1 + xi(5,2))*k 0 0] Hence, we are reduced in that case to:$ shortformdeltaprime ={0,SS,1,0,SS,a,SS,epsilon}$ where a =xi(5,2) are any complex number and epsilon=xi(6,3)=0,1$ Suppose now xi(3,1) = 0.$ xi(3,1):=0$ clear b(1,1)$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, b(1,1)**2*xi(5,2), ss, b(2,2)*xi(6,3)}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,2):=b(1,1)**2*xi(5,2)$ deltaprimemodg(6,3):=b(2,2)*xi(6,3)$ det(AUTOM):=b(2,2)**4*b(1,1)**10$ DELTAPRIMEMODADG:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [ 2 ] [0 b(1,1) *xi(5,2) 0 0 0 0] [ ] [ 2 ] [0 0 b(2,2)*xi(6,3) b(1,1) *xi(5,2) 0 0] Then, we are reduced according to whether or not xi(5,2) NEQ 0 to:$ shortformdeltaprime ={0,SS,0,0,SS,1,SS,epsilon}$ where epsilon =xi(6,3)=0,1$ or:$ shortformdeltaprime ={0,SS,0,0,SS,0,SS,1}$