The generic automorphism phi of g_{6,10} as computed by calculautom6_10.red : phi:= mat((b(1,1),0,0,0,0,0), 2 b(3,3) (b(2,1),---------,b(2,3),0,0,0), b(1,1) - b(2,3)*b(1,1) (------------------,0,b(3,3),0,0,0), b(3,3) 2 (b(4,1),b(4,2),b(4,3),b(3,3) ,b(2,3)*b(1,1),0), (b(5,1),0,b(5,3),0,b(3,3)*b(1,1),0), (b(6,1),b(6,2),b(6,3),b(4,2)*b(1,1), 2 b(4,3)*b(3,3)*b(1,1) - b(5,1)*b(3,3) - b(5,3)*b(2,3)*b(1,1) --------------------------------------------------------------, b(3,3) 2 b(3,3) *b(1,1))) 8 2 det(phi):=b(3,3) *b(1,1) generic derivation : delta:= mat((xi(1,1),0,0,0,0,0), (xi(2,1),xi(2,2),xi(2,3),0,0,0), xi(1,1) + xi(2,2) ( - xi(2,3),0,-------------------,0,0,0), 2 (xi(4,1),xi(4,2),xi(4,3),xi(1,1) + xi(2,2),xi(2,3),0), 3*xi(1,1) + xi(2,2) (xi(5,1),0,xi(5,3),0,---------------------,0), 2 (xi(6,1),xi(6,2),xi(6,3),xi(4,2), - ( - xi(4,3) + xi(5,1)), 2*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 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 0] adx2 := [ ] [-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] [ ] [0 0 0 0 0 0] adx3 := [ ] [0 0 0 0 0 0] [ ] [-1 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] adx4 := [ ] [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 0] adx5 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 -1 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(6,3):=0 phi:= mat((b(1,1),0,0,0,0,0), 2 b(3,3) (b(2,1),---------,b(2,3),0,0,0), b(1,1) - b(2,3)*b(1,1) (------------------,0,b(3,3),0,0,0), b(3,3) 2 (b(4,1),b(4,2),b(4,3),b(3,3) ,b(2,3)*b(1,1),0), (b(5,1),0,b(5,3),0,b(3,3)*b(1,1),0), (b(6,1),b(6,2),b(6,3),b(4,2)*b(1,1), - (b(5,3)*b(2,3)*b(1,1) + ( - b(4,3)*b(1,1) + b(5,1)*b(3,3))*b(3,3)) -----------------------------------------------------------------------, b(3,3) 2 b(3,3) *b(1,1))) 8 2 det(phi):=b(3,3) *b(1,1) delta:= [ 0 0 0 0 0 0] [ ] [ xi(2,1) 0 xi(2,3) 0 0 0] [ ] [ - xi(2,3) 0 0 0 0 0] [ ] [ 0 0 xi(4,3) 0 xi(2,3) 0] [ ] [ 0 0 xi(5,3) 0 0 0] [ ] [ 0 xi(6,2) 0 0 xi(4,3) 0] We denote this delta by the shortform shortformdelta:={xi(2,1), xi(2,3), ss, xi(4,3), ss, xi(5,3), ss, xi(6,2)} paramindexeslist:={{2,1},{2,3},{4,3},{5,3},{6,2}} shortformdeltaprimemodadg:={(b(3,3)**2*xi(2,1))/b(1,1)**2, (b(3,3)*xi(2,3))/b(1,1), ss, ( - ( - (b(2,3)*b(1,1)*xi(5,3) + b(4,2)*xi(2,3))*b(1,1) + ( - b(3,3)*b(1,1)*xi(4 ,3) + b(5,3)*xi(2,3))*b(3,3)))/(b(3,3)*b(1,1)), ss, b(1,1)*xi(5,3), ss, b(1,1)**2*xi(6,2)}$ deltaprimemodg(2,1):=(b(3,3)**2*xi(2,1))/b(1,1)**2$ deltaprimemodg(2,3):=(b(3,3)*xi(2,3))/b(1,1)$ deltaprimemodg(4,3):=( - ( - (b(2,3)*b(1,1)*xi(5,3) + b(4,2)*xi(2,3))*b(1,1) + ( - b(3,3)*b(1,1)*xi(4,3) + b(5,3)*xi(2,3))*b(3,3)))/(b(3,3)*b(1,1))$ deltaprimemodg(5,3):=b(1,1)*xi(5,3)$ deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$ det(AUTOM):=b(3,3)**8*b(1,1)**2$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), 2 b(3,3) *xi(2,1) b(3,3)*xi(2,3) (-----------------,0,----------------,0,0,0), 2 b(1,1) b(1,1) - b(3,3)*xi(2,3) (-------------------,0,0,0,0,0), b(1,1) (0,0,( - ( - (b(2,3)*b(1,1)*xi(5,3) + b(4,2)*xi(2,3))*b(1,1) + ( - b(3,3)*b(1,1)*xi(4,3) + b(5,3)*xi(2,3))*b(3,3)))/(b(3,3) b(3,3)*xi(2,3) *b(1,1)),0,----------------,0), b(1,1) (0,0,b(1,1)*xi(5,3),0,0,0), 2 2 (0,b(1,1) *xi(6,2),0,0,( - ( - (b(3,3) *xi(4,3) + b(4,2)*xi(2,3))*b(1,1) 2 - b(2,3)*b(1,1) *xi(5,3) + b(5,3)*b(3,3)*xi(2,3)))/(b(3,3)*b(1,1)),0 )) ************ SUBCASE 2 : we suppose xi(2,3) = 0 *********************$ xi(2,3):=0$ shortformdeltaprimemodadg:={(b(3,3)**2*xi(2,1))/b(1,1)**2, 0, ss, (b(2,3)*b(1,1)*xi(5,3) + b(3,3)**2*xi(4,3))/b(3,3), ss, b(1,1)*xi(5,3), ss, b(1,1)**2*xi(6,2)}$ deltaprimemodg(2,1):=(b(3,3)**2*xi(2,1))/b(1,1)**2$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=(b(2,3)*b(1,1)*xi(5,3) + b(3,3)**2*xi(4,3))/b(3,3)$ deltaprimemodg(5,3):=b(1,1)*xi(5,3)$ deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$ det(AUTOM):=b(3,3)**8*b(1,1)**2$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), 2 b(3,3) *xi(2,1) (-----------------,0,0,0,0,0), 2 b(1,1) (0,0,0,0,0,0), 2 b(2,3)*b(1,1)*xi(5,3) + b(3,3) *xi(4,3) (0,0,-----------------------------------------,0,0,0), b(3,3) (0,0,b(1,1)*xi(5,3),0,0,0), 2 2 b(2,3)*b(1,1)*xi(5,3) + b(3,3) *xi(4,3) (0,b(1,1) *xi(6,2),0,0,-----------------------------------------,0)) b(3,3) ************ SUBSUBCASE 2.1 : we suppose xi(5,3) NEQ 0 ********************* $ If xi(5,3) NEQ 0 we get deltaprime(5,3)=k by taking:$ b(1,1):=k/xi(5,3)$ and we get deltaprime(4,3)=0 by taking$ b(2,3):=( - b(3,3)**2*xi(4,3))/k$ shortformdeltaprimemodadg:={(b(3,3)**2*xi(5,3)**2*xi(2,1))/k**2, 0, ss, 0, ss, k, ss, (xi(6,2)*k**2)/xi(5,3)**2}$ deltaprimemodg(2,1):=(b(3,3)**2*xi(5,3)**2*xi(2,1))/k**2$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=0$ deltaprimemodg(5,3):=k$ deltaprimemodg(6,2):=(xi(6,2)*k**2)/xi(5,3)**2$ det(AUTOM):=(b(3,3)**8*k**2)/xi(5,3)**2$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 2 2 ] [ b(3,3) *xi(5,3) *xi(2,1) ] [-------------------------- 0 0 0 0 0] [ 2 ] [ k ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 k 0 0 0] [ ] [ 2 ] [ xi(6,2)*k ] [ 0 ------------ 0 0 0 0] [ 2 ] [ xi(5,3) ] If xi(6,2) NEQ 0 we get deltaprime(6,2)=k by taking:$ k:=xi(5,3)**2/xi(6,2)$ shortformdeltaprimemodadg:={(b(3,3)**2*xi(6,2)**2*xi(2,1))/xi(5,3)**2, 0, ss, 0, ss, xi(5,3)**2/xi(6,2), ss, xi(5,3)**2/xi(6,2)}$ deltaprimemodg(2,1):=(b(3,3)**2*xi(6,2)**2*xi(2,1))/xi(5,3)**2$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=0$ deltaprimemodg(5,3):=xi(5,3)**2/xi(6,2)$ deltaprimemodg(6,2):=xi(5,3)**2/xi(6,2)$ det(AUTOM):=(b(3,3)**8*xi(5,3)**2)/xi(6,2)**2$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 2 2 ] [ b(3,3) *xi(6,2) *xi(2,1) ] [-------------------------- 0 0 0 0 0] [ 2 ] [ xi(5,3) ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ xi(5,3) ] [ 0 0 ---------- 0 0 0] [ xi(6,2) ] [ ] [ 2 ] [ xi(5,3) ] [ 0 ---------- 0 0 0 0] [ xi(6,2) ] Hence, we are reduced in that case to:$ shortformdeltaprime ={epsilon,0,SS,0,SS,1,SS,eta}$ where etam =xi(6,2) =0,1 and epsilon=xi(2,1):=0,1$ ************ SUBSUBCASE 2.2 : we suppose xi(5,3) = 0 *********************$ If xi(5,3) = 0, we get :$ xi(5,3):=0$ clear b(1,1),b(2,3),k$ shortformdeltaprimemodadg:={(b(3,3)**2*xi(2,1))/b(1,1)**2, 0, ss, b(3,3)*xi(4,3), ss, 0, ss, b(1,1)**2*xi(6,2)}$ deltaprimemodg(2,1):=(b(3,3)**2*xi(2,1))/b(1,1)**2$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=b(3,3)*xi(4,3)$ deltaprimemodg(5,3):=0$ deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$ det(AUTOM):=b(3,3)**8*b(1,1)**2$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 2 ] [ b(3,3) *xi(2,1) ] [----------------- 0 0 0 0 0] [ 2 ] [ b(1,1) ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 b(3,3)*xi(4,3) 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ 0 b(1,1) *xi(6,2) 0 0 b(3,3)*xi(4,3) 0] ********* Suppose first xi(4,3) NEQ 0. *********************************$ Then we get deltaprime(4,3)=k by taking:$ b(3,3):=b(2,3)$ **** If moreover xi(6,2) NEQ 0 we get deltaprime(6,2)=k by taking:$ b(1,1):=sqrt(k/xi(6,2))$ shortformdeltaprimemodadg:={(xi(6,2)*xi(2,1)*k)/xi(4,3)**2, 0, ss, k, ss, 0, ss, k}$ deltaprimemodg(2,1):=(xi(6,2)*xi(2,1)*k)/xi(4,3)**2$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=k$ deltaprimemodg(5,3):=0$ deltaprimemodg(6,2):=k$ det(AUTOM):=k**9/(xi(6,2)*xi(4,3)**8)$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ xi(6,2)*xi(2,1)*k ] [------------------- 0 0 0 0 0] [ 2 ] [ xi(4,3) ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 k 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 k 0 0 k 0] Hence, we are reduced if that case to:$ shortformdeltaprime ={a,0,SS,1,SS,0,SS,1}$ where a =xi(2,1) is any complex$ ***** If xi(6,2) = 0 we get :$ xi(6,2):=0$ clear b(1,1)$ shortformdeltaprimemodadg:={(xi(2,1)*k**2)/(b(1,1)**2*xi(4,3)**2), 0, ss, k, ss, 0, ss, 0}$ deltaprimemodg(2,1):=(xi(2,1)*k**2)/(b(1,1)**2*xi(4,3)**2)$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=k$ deltaprimemodg(5,3):=0$ deltaprimemodg(6,2):=0$ det(AUTOM):=(b(1,1)**2*k**8)/xi(4,3)**8$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 2 ] [ xi(2,1)*k ] [------------------ 0 0 0 0 0] [ 2 2 ] [ b(1,1) *xi(4,3) ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 k 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 k 0] Then we are reduced to:$ shortformdeltaprime ={epsilon,0,SS,1,SS,0,SS,0}$ where epsilon =xi(2,1)=0,1.$ ********* Suppose now xi(4,3) = 0. *********************************$ xi(4,3):=0$ clear b(3,3),xi(6,2)$ shortformdeltaprimemodadg:={(b(3,3)**2*xi(2,1))/b(1,1)**2, 0, ss, 0, ss, 0, ss, b(1,1)**2*xi(6,2)}$ deltaprimemodg(2,1):=(b(3,3)**2*xi(2,1))/b(1,1)**2$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=0$ deltaprimemodg(5,3):=0$ deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$ det(AUTOM):=b(3,3)**8*b(1,1)**2$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 2 ] [ b(3,3) *xi(2,1) ] [----------------- 0 0 0 0 0] [ 2 ] [ b(1,1) ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ 0 b(1,1) *xi(6,2) 0 0 0 0] **** If moreover xi(6,2) NEQ 0 we get deltaprime(6,2)=k by taking:$ b(1,1):=sqrt(k/xi(6,2))$ shortformdeltaprimemodadg:={(b(3,3)**2*xi(6,2)*xi(2,1))/k, 0, ss, 0, ss, 0, ss, k}$ deltaprimemodg(2,1):=(b(3,3)**2*xi(6,2)*xi(2,1))/k$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=0$ deltaprimemodg(5,3):=0$ deltaprimemodg(6,2):=k$ det(AUTOM):=(b(3,3)**8*k)/xi(6,2)$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 2 ] [ b(3,3) *xi(6,2)*xi(2,1) ] [------------------------- 0 0 0 0 0] [ k ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 k 0 0 0 0] Hence, we are reduced if that case to:$ shortformdeltaprime ={epsilon,0,SS,0,SS,0,SS,1}$ where epsilon =xi(2,1)=0,1 $ **** If finally xi(6,2) = 0 we get :$ xi(6,2):=0$ clear b(1,1)$ shortformdeltaprimemodadg:={(b(3,3)**2*xi(2,1))/b(1,1)**2, 0, ss, 0, ss, 0, ss, 0}$ deltaprimemodg(2,1):=(b(3,3)**2*xi(2,1))/b(1,1)**2$ deltaprimemodg(2,3):=0$ deltaprimemodg(4,3):=0$ deltaprimemodg(5,3):=0$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(3,3)**8*b(1,1)**2$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 2 ] [ b(3,3) *xi(2,1) ] [----------------- 0 0 0 0 0] [ 2 ] [ b(1,1) ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] As we suppose delta NEQ 0, we have xi(2,1) neq 0 in that case:$ Hence, we are reduced if that case to:$ shortformdeltaprime ={1,0,SS,0,SS,0,SS,0}$