The generic automorphism phi of g_{6,1} as computed by calculautom6_12.red : phi:= mat((b(1,1),0,0,0,0,0), 2 (b(2,1),b(1,1) ,0,0,0,0), 3 (b(3,1),b(3,2),b(1,1) ,0,0,0), 3 (b(4,1),b(4,2),0,b(1,1) ,0,0), 2 4 (b(5,1),b(5,2), - b(2,1)*b(1,1) ,b(4,2)*b(1,1),b(1,1) ,0), 2 (b(6,1),b(6,2),b(6,3),b(5,2)*b(1,1) + b(4,2)*b(2,1) - b(4,1)*b(1,1) 2 2 5 + b(3,2)*b(2,1) - b(3,1)*b(1,1) ,b(1,1) *(b(4,2) + b(2,1)*b(1,1)),b(1,1) ) ) 18 det(phi):=b(1,1) generic derivation : delta:= mat((xi(1,1),0,0,0,0,0), (xi(2,1),2*xi(1,1),0,0,0,0), (xi(3,1),xi(3,2),3*xi(1,1),0,0,0), (xi(4,1),xi(4,2),0,3*xi(1,1),0,0), (xi(5,1),xi(5,2), - xi(2,1),xi(4,2),4*xi(1,1),0), (xi(6,1),xi(6,2),xi(6,3), - (xi(4,1) + xi(3,1) - xi(5,2)),xi(4,2) + xi(2,1), 5*xi(1,1))) [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 1 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 -1 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 as in (Cohomology tables page 50) : the eigen\ values are 0 by subtracting adjoints one then may suppose xi(4,1)=xi(4,2)=xi(5,1)=xi(6,1)=x\ i(6,2)=0 phi:= mat((b(1,1),0,0,0,0,0), 2 (b(2,1),b(1,1) ,0,0,0,0), 3 (b(3,1),b(3,2),b(1,1) ,0,0,0), 3 (b(4,1),b(4,2),0,b(1,1) ,0,0), 2 4 (b(5,1),b(5,2), - b(2,1)*b(1,1) ,b(4,2)*b(1,1),b(1,1) ,0), 2 2 (b(6,1),b(6,2),b(6,3),b(3,2)*b(2,1) - b(3,1)*b(1,1) - b(4,1)*b(1,1) 2 5 + b(4,2)*b(2,1) + b(5,2)*b(1,1),(b(4,2) + b(2,1)*b(1,1))*b(1,1) ,b(1,1) )) 18 det(phi):=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(5,2) - xi(3,1) xi(2,1) 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(1,1)*xi(2,1), ss, ( - ((b(2,1)*xi(3,2) - b(1,1)**2*xi(3,1))*b(1,1) - b(3,2)*xi(2,1)))/b(1,1), b(1,1)*xi(3,2), ss, ( - ((b(2,1)*xi(3,2) - b(1,1)**2*xi(5,2))*b(1,1) - b(3,2)*xi(2,1)))/b(1,1), ss, b(1,1)**2*xi(6,3)}$ deltaprimemodg(2,1):=b(1,1)*xi(2,1)$ deltaprimemodg(3,1):=( - ((b(2,1)*xi(3,2) - b(1,1)**2*xi(3,1))*b(1,1) - b(3,2)* xi(2,1)))/b(1,1)$ deltaprimemodg(3,2):=b(1,1)*xi(3,2)$ deltaprimemodg(5,2):=( - ((b(2,1)*xi(3,2) - b(1,1)**2*xi(5,2))*b(1,1) - b(3,2)* xi(2,1)))/b(1,1)$ deltaprimemodg(6,3):=b(1,1)**2*xi(6,3)$ det(phi):=b(1,1)**18$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(b(1,1)*xi(2,1),0,0,0,0,0),(( - ((b(2,1)*xi(3,2) - b(1,1)**2* xi(3,1))*b(1,1) - b(3,2)*xi(2,1)))/b(1,1),b(1,1)*xi(3,2),0,0,0,0),(0,0,0,0,0,0), (0,( - ((b(2,1)*xi(3,2) - b(1,1)**2*xi(5,2))*b(1,1) - b(3,2)*xi(2,1)))/b(1,1), - b(1,1)*xi(2,1),0,0,0),(0,0,b(1,1)**2*xi(6,3),(xi(5,2) - xi(3,1))*b(1,1)**2,b(1, 1)*xi(2,1),0))$ ********* We suppose here that xi(2,1) = 0.$ xi(2,1):=0$ shortformdeltaprimemodadg:={0, ss, - (b(2,1)*xi(3,2) - b(1,1)**2*xi(3,1)), b(1,1)*xi(3,2), ss, - (b(2,1)*xi(3,2) - b(1,1)**2*xi(5,2)), ss, b(1,1)**2*xi(6,3)}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):= - (b(2,1)*xi(3,2) - b(1,1)**2*xi(3,1))$ deltaprimemodg(3,2):=b(1,1)*xi(3,2)$ deltaprimemodg(5,2):= - (b(2,1)*xi(3,2) - b(1,1)**2*xi(5,2))$ deltaprimemodg(6,3):=b(1,1)**2*xi(6,3)$ det(phi):=b(1,1)**18$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),( - (b(2,1)*xi(3,2) - b(1,1)**2*xi(3,1)),b(1,1)* xi(3,2),0,0,0,0),(0,0,0,0,0,0),(0, - (b(2,1)*xi(3,2) - b(1,1)**2*xi(5,2)),0,0,0, 0),(0,0,b(1,1)**2*xi(6,3),(xi(5,2) - xi(3,1))*b(1,1)**2,0,0))$ ********SUBSUBCASE I:$ Suppose xi(3,2) neq 0$ Then one may suppose xi(3,2):=1:$ xi(3,2):=1$ Then one keeps deltaprimemodg(3,2)=k $ and deltaprimemodg(3,1)=0 by taking:$ b(1,1):=k$ b(2,1):=xi(3,1)*k**2$ shortformdeltaprimemodadg:={0, ss, 0, k, ss, (xi(5,2) - xi(3,1))*k**2, ss, xi(6,3)*k**2}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=k$ deltaprimemodg(5,2):=(xi(5,2) - xi(3,1))*k**2$ deltaprimemodg(6,3):=xi(6,3)*k**2$ det(phi):=k**18$ 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),(0,(xi(5,2) - xi(3,1 ))*k**2,0,0,0,0),(0,0,xi(6,3)*k**2,(xi(5,2) - xi(3,1))*k**2,0,0))$ shortformdeltaprimemodadg:={0, ss, 0, k, ss, (xi(5,2) - xi(3,1))*k**2, ss, xi(6,3)*k**2}$ Suppose xi(6,3) neq 0 .$ Then we are reduced to:$ shortformdeltaprime ={0,SS,0,1,SS,a,SS,1}$ where a is any complex$ Suppose xi(6,3) = 0 .$ shortformdeltaprimemodadg:={0, ss, 0, k, ss, (xi(5,2) - xi(3,1))*k**2, ss, 0}$ Then we are reduced to:$ shortformdeltaprime ={0,SS,0,1,SS,epsilon,SS,0}$ clear xi(6,3),b(2,1),b(1,1)$ ********SUBSUBCASE II:$ Suppose xi(3,2) :=0$ xi(3,2):=0$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(3,1), 0, ss, b(1,1)**2*xi(5,2), ss, b(1,1)**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(1,1)**2*xi(6,3)$ det(phi):=b(1,1)**18$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(b(1,1)**2*xi(3,1),0,0,0,0,0),(0,0,0,0,0,0),(0,b (1,1)**2*xi(5,2),0,0,0,0),(0,0,b(1,1)**2*xi(6,3),(xi(5,2) - xi(3,1))*b(1,1)**2,0 ,0))$ Suppose xi(3,1) neq 0 .$ Then one may suppose xi(3,1):=1:$ xi(3,1):=1$ Then one keeps deltaprimemodg(3,1)=k by taking:$ b(1,1):=sqrt(k)$ shortformdeltaprimemodadg:={0, ss, k, 0, ss, xi(5,2)*k, ss, xi(6,3)*k}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=k$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,2):=xi(5,2)*k$ deltaprimemodg(6,3):=xi(6,3)*k$ det(phi):=k**9$ DELTAPRIMEMODADG:= mat((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,xi(6,3)*k,(xi(5,2) - 1)*k,0,0))$ Then we are reduced to:$ shortformdeltaprime ={0,SS,1,0,SS,a,SS,b}$ where a,b are any complexes$ Suppose now xi(3,1) = 0 .$ xi(3,1):=0$ shortformdeltaprimemodadg:={0, ss, 0, 0, ss, xi(5,2)*k, ss, xi(6,3)*k}$ deltaprimemodg(2,1):=0$ deltaprimemodg(3,1):=0$ deltaprimemodg(3,2):=0$ deltaprimemodg(5,2):=xi(5,2)*k$ deltaprimemodg(6,3):=xi(6,3)*k$ det(phi):=k**9$ DELTAPRIMEMODADG:= mat((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,xi(5,2)*k,0,0,0,0 ),(0,0,xi(6,3)*k,xi(5,2)*k,0,0))$ Then, according to whether or not xi(5,2) vanishes, we are reduced to either:$ shortformdeltaprime ={0,SS,0,0,SS,1,SS,a}$ where a is any complex$ or : shortformdeltaprime ={0,SS,0,0,SS,0,SS,1}$