CASE2N00 The generic automorphism phi of n x C**3 : phi:= [b(1,1) b(1,2) 0 0 0 0 ] [ ] [b(2,1) b(2,2) 0 0 0 0 ] [ ] [b(3,1) b(3,2) - b(2,1)*b(1,2) + b(2,2)*b(1,1) b(3,4) b(3,5) b(3,6)] [ ] [b(4,1) b(4,2) 0 b(4,4) b(4,5) b(4,6)] [ ] [b(5,1) b(5,2) 0 b(5,4) b(5,5) b(5,6)] [ ] [b(6,1) b(6,2) 0 b(6,4) b(6,5) b(6,6)] det(phi):=(( - b(5,4)*b(4,5) + b(5,5)*b(4,4))*b(6,6) - ( - b(5,4)*b(4,6) + b(5,6)*b(4,4))*b(6,5) + ( - b(5,5)*b(4,6) + b(5,6)*b(4,5))*b(6,4)) 2 *( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) generic derivation : delta:= [xi(1,1) xi(1,2) 0 0 0 0 ] [ ] [xi(2,1) xi(2,2) 0 0 0 0 ] [ ] [xi(3,1) xi(3,2) xi(1,1) + xi(2,2) xi(3,4) xi(3,5) xi(3,6)] [ ] [xi(4,1) xi(4,2) 0 xi(4,4) xi(4,5) xi(4,6)] [ ] [xi(5,1) xi(5,2) 0 xi(5,4) xi(5,5) xi(5,6)] [ ] [xi(6,1) xi(6,2) 0 xi(6,4) xi(6,5) xi(6,6)] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] adx1 := [ ] [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] [ ] [-1 0 0 0 0 0] adx2 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : The matrices MATA:=((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2))) is nilpotent And MATB:=((xi(4,4),xi(4,5),xi(4,6)),(xi(5,4),xi(5,5),xi(5,6)),(xi(6,4),xi(6,\ 5),xi(6,6))) are nilpotent (WE denote MATA,MATB, since B is already for the entries of phi) **We consider here the case 2 where MATA = 0 and MATB has nilpotent ordre 1.*** xi(1,1):=0,xi(1,2):=0,xi(2,1):=0,xi(2,2):=0 MATB:= [0 0 0] [ ] [0 0 0] [ ] [0 0 0] MATC:= [xi(3,4)] [ ] [xi(3,5)] [ ] [xi(3,6)] MATD:= [xi(4,1) xi(4,2)] [ ] [xi(5,1) xi(5,2)] [ ] [xi(6,1) xi(6,2)] In block matrices, we get : phi:=mat((MATU,0,0),(MATR,det(MATU),MATS),(MATT,0,MATV)) where MATU is in GL(2), MATV in GL(3). delta:=mat((MATA,0,0),(0,0,MATC),(MATD,0,MATB)) Under the action of phi one gets: phi**(-1)*delta*phi:=mat((MATU**(-1)*MATA*MATU,0,0),(**,0,MATCprime),(MATDprim\ e,0,MATV**(-1)*MATB*MATV)) Hence we may suppose for MATB : MATB:= [0 1 0] [ ] [0 0 0] [ ] [0 0 0] by subtracting adjoints one then may suppose: xi(3,1):=0,xi(3,2):=0. delta:= [ 0 0 0 0 0 0 ] [ ] [ 0 0 0 0 0 0 ] [ ] [ 0 0 0 xi(3,4) xi(3,5) xi(3,6)] [ ] [xi(4,1) xi(4,2) 0 0 1 0 ] [ ] [xi(5,1) xi(5,2) 0 0 0 0 ] [ ] [xi(6,1) xi(6,2) 0 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(4,1), xi(4,2), ss, xi(5,1), xi(5,2), ss, xi(6,1), xi(6,2), ss, xi(3,4), xi(3,5), xi(3,6)} paramindexeslist:={{4,1},{4,2},{5,1},{5,2},{6,1},{6,2},{3,4},{3,5},{3,6}} to keep in MATB deltaprimemodadg(4,5):=k and the others zero, one has to take :$ MATV:= mat((b(5,5)*k,b(4,5),b(4,6)),(0,b(5,5),0),(0,b(6,5),b(6,6)))$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={( - b(4,5)*b(2,1)*xi(5,2) + b(4,5)*b(2,2)*xi(5,1) - b(4,6)*b(2,1)*xi(6,2) + b(4,6)*b(2,2)*xi(6,1) - b(5,1)*b(2,2)*k + b(5,2)*b(2,1)* k - b(5,5)*b(2,1)*xi(4,2)*k + b(5,5)*b(2,2)*xi(4,1)*k)/( - b(2,1)*b(1,2) + b(2,2 )*b(1,1)), (b(4,5)*b(1,1)*xi(5,2) - b(4,5)*b(1,2)*xi(5,1) + b(4,6)*b(1,1)*xi(6,2) - b(4,6)* b(1,2)*xi(6,1) + b(5,1)*b(1,2)*k - b(5,2)*b(1,1)*k + b(5,5)*b(1,1)*xi(4,2)*k - b (5,5)*b(1,2)*xi(4,1)*k)/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ss, (b(5,5)*( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) , (b(5,5)*(b(1,1)*xi(5,2) - b(1,2)*xi(5,1)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ss, ( - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2,2)*xi(5,1) - b(6,6)*b(2,1)*xi(6,2) + b(6, 6)*b(2,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), (b(6,5)*b(1,1)*xi(5,2) - b(6,5)*b(1,2)*xi(5,1) + b(6,6)*b(1,1)*xi(6,2) - b(6,6)* b(1,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ss, (xi(3,4)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(5,5)*k), ( - b(6,5)*b(4,6)*b(2,1)*b(1,2)*xi(3,4) + b(6,5)*b(4,6)*b(2,2)*b(1,1)*xi(3,4) + b(6,5)*b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k - b(6,5)*b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k + b(6,6)*b(4,5)*b(2,1)*b(1,2)*xi(3,4) - b(6,6)*b(4,5)*b(2,2)*b(1,1)*xi(3,4) - b(6, 6)*b(5,5)*b(2,1)*b(1,2)*xi(3,5)*k + b(6,6)*b(5,5)*b(2,2)*b(1,1)*xi(3,5)*k + b(6, 6)*b(5,5)*b(3,4)*k)/(b(6,6)*b(5,5)**2*k), (b(4,6)*b(2,1)*b(1,2)*xi(3,4) - b(4,6)*b(2,2)*b(1,1)*xi(3,4) - b(5,5)*b(2,1)*b(1 ,2)*xi(3,6)*k + b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k)/(b(6,6)*b(5,5)*k)}$ deltaprimemodg(4,1):=( - b(4,5)*b(2,1)*xi(5,2) + b(4,5)*b(2,2)*xi(5,1) - b(4,6)* b(2,1)*xi(6,2) + b(4,6)*b(2,2)*xi(6,1) - b(5,1)*b(2,2)*k + b(5,2)*b(2,1)*k - b(5 ,5)*b(2,1)*xi(4,2)*k + b(5,5)*b(2,2)*xi(4,1)*k)/( - b(2,1)*b(1,2) + b(2,2)*b(1,1 ))$ deltaprimemodg(4,2):=(b(4,5)*b(1,1)*xi(5,2) - b(4,5)*b(1,2)*xi(5,1) + b(4,6)*b(1 ,1)*xi(6,2) - b(4,6)*b(1,2)*xi(6,1) + b(5,1)*b(1,2)*k - b(5,2)*b(1,1)*k + b(5,5) *b(1,1)*xi(4,2)*k - b(5,5)*b(1,2)*xi(4,1)*k)/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(5,1):=(b(5,5)*( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1)))/( - b(2,1)*b( 1,2) + b(2,2)*b(1,1))$ deltaprimemodg(5,2):=(b(5,5)*(b(1,1)*xi(5,2) - b(1,2)*xi(5,1)))/( - b(2,1)*b(1,2 ) + b(2,2)*b(1,1))$ deltaprimemodg(6,1):=( - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2,2)*xi(5,1) - b(6,6)* b(2,1)*xi(6,2) + b(6,6)*b(2,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,2):=(b(6,5)*b(1,1)*xi(5,2) - b(6,5)*b(1,2)*xi(5,1) + b(6,6)*b(1 ,1)*xi(6,2) - b(6,6)*b(1,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(3,4):=(xi(3,4)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(5,5)*k)$ deltaprimemodg(3,5):=( - b(6,5)*b(4,6)*b(2,1)*b(1,2)*xi(3,4) + b(6,5)*b(4,6)*b(2 ,2)*b(1,1)*xi(3,4) + b(6,5)*b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k - b(6,5)*b(5,5)*b(2,2 )*b(1,1)*xi(3,6)*k + b(6,6)*b(4,5)*b(2,1)*b(1,2)*xi(3,4) - b(6,6)*b(4,5)*b(2,2)* b(1,1)*xi(3,4) - b(6,6)*b(5,5)*b(2,1)*b(1,2)*xi(3,5)*k + b(6,6)*b(5,5)*b(2,2)*b( 1,1)*xi(3,5)*k + b(6,6)*b(5,5)*b(3,4)*k)/(b(6,6)*b(5,5)**2*k)$ deltaprimemodg(3,6):=(b(4,6)*b(2,1)*b(1,2)*xi(3,4) - b(4,6)*b(2,2)*b(1,1)*xi(3,4 ) - b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k + b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k)/(b(6,6)*b(5 ,5)*k)$ det(AUTOM):=b(6,6)*b(5,5)**2*k*(b(2,1)**2*b(1,2)**2 - 2*b(2,2)*b(2,1)*b(1,2)*b(1 ,1) + b(2,2)**2*b(1,1)**2)$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), xi(3,4)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) (0,0,0,--------------------------------------------,( b(5,5)*k - b(6,5)*b(4,6)*b(2,1)*b(1,2)*xi(3,4) + b(6,5)*b(4,6)*b(2,2)*b(1,1)*xi(3,4) + b(6,5)*b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k - b(6,5)*b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k + b(6,6)*b(4,5)*b(2,1)*b(1,2)*xi(3,4) - b(6,6)*b(4,5)*b(2,2)*b(1,1)*xi(3,4) - b(6,6)*b(5,5)*b(2,1)*b(1,2)*xi(3,5)*k + b(6,6)*b(5,5)*b(2,2)*b(1,1)*xi(3,5)*k + b(6,6)*b(5,5)*b(3,4)*k)/( 2 b(6,6)*b(5,5) *k),(b(4,6)*b(2,1)*b(1,2)*xi(3,4) - b(4,6)*b(2,2)*b(1,1)*xi(3,4) - b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k + b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k)/(b(6,6)*b(5,5)*k)), (( - b(4,5)*b(2,1)*xi(5,2) + b(4,5)*b(2,2)*xi(5,1) - b(4,6)*b(2,1)*xi(6,2) + b(4,6)*b(2,2)*xi(6,1) - b(5,1)*b(2,2)*k + b(5,2)*b(2,1)*k - b(5,5)*b(2,1)*xi(4,2)*k + b(5,5)*b(2,2)*xi(4,1)*k)/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),(b(4,5)*b(1,1)*xi(5,2) - b(4,5)*b(1,2)*xi(5,1) + b(4,6)*b(1,1)*xi(6,2) - b(4,6)*b(1,2)*xi(6,1) + b(5,1)*b(1,2)*k - b(5,2)*b(1,1)*k + b(5,5)*b(1,1)*xi(4,2)*k - b(5,5)*b(1,2)*xi(4,1)*k)/ ( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),0,0,k,0), b(5,5)*( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1)) (---------------------------------------------, - b(2,1)*b(1,2) + b(2,2)*b(1,1) b(5,5)*(b(1,1)*xi(5,2) - b(1,2)*xi(5,1)) ------------------------------------------,0,0,0,0), - b(2,1)*b(1,2) + b(2,2)*b(1,1) (( - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2,2)*xi(5,1) - b(6,6)*b(2,1)*xi(6,2) + b(6,6)*b(2,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),( b(6,5)*b(1,1)*xi(5,2) - b(6,5)*b(1,2)*xi(5,1) + b(6,6)*b(1,1)*xi(6,2) - b(6,6)*b(1,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),0,0,0,0)) One gets deltaprime(4,1):=0 if we take :$ b(5,1):=( - b(4,5)*b(2,1)*xi(5,2) + b(4,5)*b(2,2)*xi(5,1) - b(4,6)*b(2,1)*xi(6,2 ) + b(4,6)*b(2,2)*xi(6,1) + b(5,2)*b(2,1)*k - b(5,5)*b(2,1)*xi(4,2)*k + b(5,5)*b (2,2)*xi(4,1)*k)/(b(2,2)*k)$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, (b(4,5)*xi(5,2) + b(4,6)*xi(6,2) - b(5,2)*k + b(5,5)*xi(4,2)*k)/b(2,2), ss, (b(5,5)*( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) , (b(5,5)*(b(1,1)*xi(5,2) - b(1,2)*xi(5,1)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ss, ( - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2,2)*xi(5,1) - b(6,6)*b(2,1)*xi(6,2) + b(6, 6)*b(2,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), (b(6,5)*b(1,1)*xi(5,2) - b(6,5)*b(1,2)*xi(5,1) + b(6,6)*b(1,1)*xi(6,2) - b(6,6)* b(1,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ss, (xi(3,4)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(5,5)*k), ( - b(6,5)*b(4,6)*b(2,1)*b(1,2)*xi(3,4) + b(6,5)*b(4,6)*b(2,2)*b(1,1)*xi(3,4) + b(6,5)*b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k - b(6,5)*b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k + b(6,6)*b(4,5)*b(2,1)*b(1,2)*xi(3,4) - b(6,6)*b(4,5)*b(2,2)*b(1,1)*xi(3,4) - b(6, 6)*b(5,5)*b(2,1)*b(1,2)*xi(3,5)*k + b(6,6)*b(5,5)*b(2,2)*b(1,1)*xi(3,5)*k + b(6, 6)*b(5,5)*b(3,4)*k)/(b(6,6)*b(5,5)**2*k), (b(4,6)*b(2,1)*b(1,2)*xi(3,4) - b(4,6)*b(2,2)*b(1,1)*xi(3,4) - b(5,5)*b(2,1)*b(1 ,2)*xi(3,6)*k + b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k)/(b(6,6)*b(5,5)*k)}$ deltaprimemodg(4,1):=0$ deltaprimemodg(4,2):=(b(4,5)*xi(5,2) + b(4,6)*xi(6,2) - b(5,2)*k + b(5,5)*xi(4,2 )*k)/b(2,2)$ deltaprimemodg(5,1):=(b(5,5)*( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1)))/( - b(2,1)*b( 1,2) + b(2,2)*b(1,1))$ deltaprimemodg(5,2):=(b(5,5)*(b(1,1)*xi(5,2) - b(1,2)*xi(5,1)))/( - b(2,1)*b(1,2 ) + b(2,2)*b(1,1))$ deltaprimemodg(6,1):=( - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2,2)*xi(5,1) - b(6,6)* b(2,1)*xi(6,2) + b(6,6)*b(2,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,2):=(b(6,5)*b(1,1)*xi(5,2) - b(6,5)*b(1,2)*xi(5,1) + b(6,6)*b(1 ,1)*xi(6,2) - b(6,6)*b(1,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(3,4):=(xi(3,4)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(5,5)*k)$ deltaprimemodg(3,5):=( - b(6,5)*b(4,6)*b(2,1)*b(1,2)*xi(3,4) + b(6,5)*b(4,6)*b(2 ,2)*b(1,1)*xi(3,4) + b(6,5)*b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k - b(6,5)*b(5,5)*b(2,2 )*b(1,1)*xi(3,6)*k + b(6,6)*b(4,5)*b(2,1)*b(1,2)*xi(3,4) - b(6,6)*b(4,5)*b(2,2)* b(1,1)*xi(3,4) - b(6,6)*b(5,5)*b(2,1)*b(1,2)*xi(3,5)*k + b(6,6)*b(5,5)*b(2,2)*b( 1,1)*xi(3,5)*k + b(6,6)*b(5,5)*b(3,4)*k)/(b(6,6)*b(5,5)**2*k)$ deltaprimemodg(3,6):=(b(4,6)*b(2,1)*b(1,2)*xi(3,4) - b(4,6)*b(2,2)*b(1,1)*xi(3,4 ) - b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k + b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k)/(b(6,6)*b(5 ,5)*k)$ det(AUTOM):=b(6,6)*b(5,5)**2*k*(b(2,1)**2*b(1,2)**2 - 2*b(2,2)*b(2,1)*b(1,2)*b(1 ,1) + b(2,2)**2*b(1,1)**2)$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), xi(3,4)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) (0,0,0,--------------------------------------------,( b(5,5)*k - b(6,5)*b(4,6)*b(2,1)*b(1,2)*xi(3,4) + b(6,5)*b(4,6)*b(2,2)*b(1,1)*xi(3,4) + b(6,5)*b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k - b(6,5)*b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k + b(6,6)*b(4,5)*b(2,1)*b(1,2)*xi(3,4) - b(6,6)*b(4,5)*b(2,2)*b(1,1)*xi(3,4) - b(6,6)*b(5,5)*b(2,1)*b(1,2)*xi(3,5)*k + b(6,6)*b(5,5)*b(2,2)*b(1,1)*xi(3,5)*k + b(6,6)*b(5,5)*b(3,4)*k)/( 2 b(6,6)*b(5,5) *k),(b(4,6)*b(2,1)*b(1,2)*xi(3,4) - b(4,6)*b(2,2)*b(1,1)*xi(3,4) - b(5,5)*b(2,1)*b(1,2)*xi(3,6)*k + b(5,5)*b(2,2)*b(1,1)*xi(3,6)*k)/(b(6,6)*b(5,5)*k)), b(4,5)*xi(5,2) + b(4,6)*xi(6,2) - b(5,2)*k + b(5,5)*xi(4,2)*k (0,---------------------------------------------------------------,0,0,k,0), b(2,2) b(5,5)*( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1)) (---------------------------------------------, - b(2,1)*b(1,2) + b(2,2)*b(1,1) b(5,5)*(b(1,1)*xi(5,2) - b(1,2)*xi(5,1)) ------------------------------------------,0,0,0,0), - b(2,1)*b(1,2) + b(2,2)*b(1,1) (( - b(6,5)*b(2,1)*xi(5,2) + b(6,5)*b(2,2)*xi(5,1) - b(6,6)*b(2,1)*xi(6,2) + b(6,6)*b(2,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),( b(6,5)*b(1,1)*xi(5,2) - b(6,5)*b(1,2)*xi(5,1) + b(6,6)*b(1,1)*xi(6,2) - b(6,6)*b(1,2)*xi(6,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),0,0,0,0)) One then gets deltaprime(4,2):=0 if we take :$ b(5,2):=(b(4,5)*xi(5,2) + b(4,6)*xi(6,2) + b(5,5)*xi(4,2)*k)/k$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, 0, ss, (( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(5,5))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) , ( - ( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(5,5))/( - b(2,1)*b(1,2) + b(2,2)*b(1, 1)), ss, (( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(6,5) + ( - b(2,1)*xi(6,2) + b(2,2)*xi(6, 1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ( - (( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(6,5) + ( - b(1,1)*xi(6,2) + b(1,2)* xi(6,1))*b(6,6)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ss, (( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,4))/(b(5,5)*k), ( - ( - b(6,6)*b(5,5)*b(3,4)*k + ( - ( - b(6,5)*xi(3,6) + b(6,6)*xi(3,5))*b(5,5) *k + ( - b(6,5)*b(4,6) + b(6,6)*b(4,5))*xi(3,4))*( - b(2,1)*b(1,2) + b(2,2)*b(1, 1))))/(b(6,6)*b(5,5)**2*k), (( - b(4,6)*xi(3,4) + b(5,5)*xi(3,6)*k)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(6 ,6)*b(5,5)*k)}$ deltaprimemodg(4,1):=0$ deltaprimemodg(4,2):=0$ deltaprimemodg(5,1):=(( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(5,5))/( - b(2,1)*b( 1,2) + b(2,2)*b(1,1))$ deltaprimemodg(5,2):=( - ( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(5,5))/( - b(2,1) *b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,1):=(( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(6,5) + ( - b(2,1)* xi(6,2) + b(2,2)*xi(6,1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,2):=( - (( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(6,5) + ( - b(1, 1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(3,4):=(( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,4))/(b(5,5)*k)$ deltaprimemodg(3,5):=( - ( - b(6,6)*b(5,5)*b(3,4)*k + ( - ( - b(6,5)*xi(3,6) + b (6,6)*xi(3,5))*b(5,5)*k + ( - b(6,5)*b(4,6) + b(6,6)*b(4,5))*xi(3,4))*( - b(2,1) *b(1,2) + b(2,2)*b(1,1))))/(b(6,6)*b(5,5)**2*k)$ deltaprimemodg(3,6):=(( - b(4,6)*xi(3,4) + b(5,5)*xi(3,6)*k)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(6,6)*b(5,5)*k)$ det(AUTOM):=( - b(2,1)*b(1,2) + b(2,2)*b(1,1))**2*b(6,6)*b(5,5)**2*k$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), ( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,4) (0,0,0,--------------------------------------------,( - ( b(5,5)*k - b(6,6)*b(5,5)*b(3,4)*k + ( - ( - b(6,5)*xi(3,6) + b(6,6)*xi(3,5))*b(5,5)*k + ( - b(6,5)*b(4,6) + b(6,6)*b(4,5))*xi(3,4)) 2 *( - b(2,1)*b(1,2) + b(2,2)*b(1,1))))/(b(6,6)*b(5,5) *k), ( - b(4,6)*xi(3,4) + b(5,5)*xi(3,6)*k)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) --------------------------------------------------------------------------- b(6,6)*b(5,5)*k ), (0,0,0,0,k,0), ( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(5,5) (---------------------------------------------, - b(2,1)*b(1,2) + b(2,2)*b(1,1) - ( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(5,5) ------------------------------------------------,0,0,0,0), - b(2,1)*b(1,2) + b(2,2)*b(1,1) ((( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(6,5) + ( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),( - (( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(6,5) + ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),0,0,0,0)) Hence we may suppose xi(4,1):=0 and xi(4,2):=0,$ and we keep that by the above choices.$ xi(4,1):=0$ xi(4,2):=0$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, 0, ss, (( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(5,5))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) , ( - ( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(5,5))/( - b(2,1)*b(1,2) + b(2,2)*b(1, 1)), ss, (( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(6,5) + ( - b(2,1)*xi(6,2) + b(2,2)*xi(6, 1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ( - (( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(6,5) + ( - b(1,1)*xi(6,2) + b(1,2)* xi(6,1))*b(6,6)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ss, (( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,4))/(b(5,5)*k), ( - ( - b(6,6)*b(5,5)*b(3,4)*k + ( - ( - b(6,5)*xi(3,6) + b(6,6)*xi(3,5))*b(5,5) *k + ( - b(6,5)*b(4,6) + b(6,6)*b(4,5))*xi(3,4))*( - b(2,1)*b(1,2) + b(2,2)*b(1, 1))))/(b(6,6)*b(5,5)**2*k), (( - b(4,6)*xi(3,4) + b(5,5)*xi(3,6)*k)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(6 ,6)*b(5,5)*k)}$ deltaprimemodg(4,1):=0$ deltaprimemodg(4,2):=0$ deltaprimemodg(5,1):=(( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(5,5))/( - b(2,1)*b( 1,2) + b(2,2)*b(1,1))$ deltaprimemodg(5,2):=( - ( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(5,5))/( - b(2,1) *b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,1):=(( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(6,5) + ( - b(2,1)* xi(6,2) + b(2,2)*xi(6,1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,2):=( - (( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(6,5) + ( - b(1, 1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(3,4):=(( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,4))/(b(5,5)*k)$ deltaprimemodg(3,5):=( - ( - b(6,6)*b(5,5)*b(3,4)*k + ( - ( - b(6,5)*xi(3,6) + b (6,6)*xi(3,5))*b(5,5)*k + ( - b(6,5)*b(4,6) + b(6,6)*b(4,5))*xi(3,4))*( - b(2,1) *b(1,2) + b(2,2)*b(1,1))))/(b(6,6)*b(5,5)**2*k)$ deltaprimemodg(3,6):=(( - b(4,6)*xi(3,4) + b(5,5)*xi(3,6)*k)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(6,6)*b(5,5)*k)$ det(AUTOM):=( - b(2,1)*b(1,2) + b(2,2)*b(1,1))**2*b(6,6)*b(5,5)**2*k$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), ( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,4) (0,0,0,--------------------------------------------,( - ( b(5,5)*k - b(6,6)*b(5,5)*b(3,4)*k + ( - ( - b(6,5)*xi(3,6) + b(6,6)*xi(3,5))*b(5,5)*k + ( - b(6,5)*b(4,6) + b(6,6)*b(4,5))*xi(3,4)) 2 *( - b(2,1)*b(1,2) + b(2,2)*b(1,1))))/(b(6,6)*b(5,5) *k), ( - b(4,6)*xi(3,4) + b(5,5)*xi(3,6)*k)*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) --------------------------------------------------------------------------- b(6,6)*b(5,5)*k ), (0,0,0,0,k,0), ( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(5,5) (---------------------------------------------, - b(2,1)*b(1,2) + b(2,2)*b(1,1) - ( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(5,5) ------------------------------------------------,0,0,0,0), - b(2,1)*b(1,2) + b(2,2)*b(1,1) ((( - b(2,1)*xi(5,2) + b(2,2)*xi(5,1))*b(6,5) + ( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),( - (( - b(1,1)*xi(5,2) + b(1,2)*xi(5,1))*b(6,5) + ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),0,0,0,0)) Now, suppose xi(3,4) :=0.$ xi(3,4):=0$ ******* Suppose now that xi(5,1) xi(5,2) are both zero .$ xi(5,1):=0, xi(5,2):=0$ xi(5,1):=0$ xi(5,2):=0$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, 0, ss, 0, 0, ss, (( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) , ( - ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1, 1)), ss, 0, (b(6,6)*b(3,4) + ( - b(6,5)*xi(3,6) + b(6,6)*xi(3,5))*( - b(2,1)*b(1,2) + b(2,2) *b(1,1)))/(b(6,6)*b(5,5)), (( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6))/b(6,6)}$ deltaprimemodg(4,1):=0$ deltaprimemodg(4,2):=0$ deltaprimemodg(5,1):=0$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=(( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6))/( - b(2,1)*b( 1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,2):=( - ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6))/( - b(2,1) *b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(3,4):=0$ deltaprimemodg(3,5):=(b(6,6)*b(3,4) + ( - b(6,5)*xi(3,6) + b(6,6)*xi(3,5))*( - b (2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(6,6)*b(5,5))$ deltaprimemodg(3,6):=(( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6))/b(6,6)$ det(AUTOM):=( - b(2,1)*b(1,2) + b(2,2)*b(1,1))**2*b(6,6)*b(5,5)**2*k$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,(b(6,6)*b(3,4) + ( - b(6,5)*xi(3,6) + b(6,6)*xi(3,5)) *( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/(b(6,6)*b(5,5)), ( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6) --------------------------------------------), b(6,6) (0,0,0,0,k,0), (0,0,0,0,0,0), ( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6) (---------------------------------------------, - b(2,1)*b(1,2) + b(2,2)*b(1,1) - ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6) ------------------------------------------------,0,0,0,0)) - b(2,1)*b(1,2) + b(2,2)*b(1,1) Then we get deltaprimemodg(3,5):=0 by taking :$ b(3,4):=( - ( - b(6,5)*xi(3,6) + b(6,6)*xi(3,5))*( - b(2,1)*b(1,2) + b(2,2)*b(1, 1)))/b(6,6)$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, 0, ss, 0, 0, ss, (( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) , ( - ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6))/( - b(2,1)*b(1,2) + b(2,2)*b(1, 1)), ss, 0, 0, (( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6))/b(6,6)}$ deltaprimemodg(4,1):=0$ deltaprimemodg(4,2):=0$ deltaprimemodg(5,1):=0$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=(( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6))/( - b(2,1)*b( 1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,2):=( - ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6))/( - b(2,1) *b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(3,4):=0$ deltaprimemodg(3,5):=0$ deltaprimemodg(3,6):=(( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6))/b(6,6)$ det(AUTOM):=( - b(2,1)*b(1,2) + b(2,2)*b(1,1))**2*b(6,6)*b(5,5)**2*k$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), ( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6) (0,0,0,0,0,--------------------------------------------), b(6,6) (0,0,0,0,k,0), (0,0,0,0,0,0), ( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1))*b(6,6) (---------------------------------------------, - b(2,1)*b(1,2) + b(2,2)*b(1,1) - ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(6,6) ------------------------------------------------,0,0,0,0)) - b(2,1)*b(1,2) + b(2,2)*b(1,1) ************** SUBCASE 1 : xi(6,1) and xi(6,2) both zero ******************$ xi(6,1):=0$ xi(6,2):=0$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, 0, ss, 0, 0, ss, 0, 0, ss, 0, 0, (( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6))/b(6,6)}$ deltaprimemodg(4,1):=0$ deltaprimemodg(4,2):=0$ deltaprimemodg(5,1):=0$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=0$ deltaprimemodg(3,4):=0$ deltaprimemodg(3,5):=0$ deltaprimemodg(3,6):=(( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6))/b(6,6)$ det(AUTOM):=( - b(2,1)*b(1,2) + b(2,2)*b(1,1))**2*b(6,6)*b(5,5)**2*k$ DELTAPRIMEMODADG:= [0 0 0 0 0 0 ] [ ] [0 0 0 0 0 0 ] [ ] [ ( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6) ] [0 0 0 0 0 --------------------------------------------] [ b(6,6) ] [ ] [0 0 0 0 k 0 ] [ ] [0 0 0 0 0 0 ] [ ] [0 0 0 0 0 0 ] If xi(3,6) = 0, we get for gtildedelta a direct product by C x(6) $ As we dismiss direct products, we may suppose xi(3,6) neq 0.$ We are reduced in that case to : $ (0,0;0,0;0,0;0,0,1)$ ************** SUBCASE 2 : xi(6,1) and xi(6,2) not both zero **************** $ In that case, by suitable choice of b(1,1),b(1,2),b(2,1),b(2,2), one may$ suppose:$ xi(6,1):=0$ xi(6,2):=1$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, 0, ss, 0, 0, ss, ( - b(6,6)*b(2,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), (b(6,6)*b(1,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ss, 0, 0, (( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6))/b(6,6)}$ deltaprimemodg(4,1):=0$ deltaprimemodg(4,2):=0$ deltaprimemodg(5,1):=0$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=( - b(6,6)*b(2,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,2):=(b(6,6)*b(1,1))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(3,4):=0$ deltaprimemodg(3,5):=0$ deltaprimemodg(3,6):=(( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6))/b(6,6)$ det(AUTOM):=( - b(2,1)*b(1,2) + b(2,2)*b(1,1))**2*b(6,6)*b(5,5)**2*k$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), ( - b(2,1)*b(1,2) + b(2,2)*b(1,1))*xi(3,6) (0,0,0,0,0,--------------------------------------------), b(6,6) (0,0,0,0,k,0), (0,0,0,0,0,0), - b(6,6)*b(2,1) b(6,6)*b(1,1) (----------------------------------,----------------------------------,0,0,0 - b(2,1)*b(1,2) + b(2,2)*b(1,1) - b(2,1)*b(1,2) + b(2,2)*b(1,1) ,0)) and we keep deltaprimemodg(6,2):=k and deltaprimemodg(6,1):=0 by taking :$ b(2,1):=0$ b(2,2):=b(6,6)/k$ With the generic automorphism one gets$ shortformdeltaprimemodadg:={0, 0, ss, 0, 0, ss, 0, k, ss, 0, 0, (b(1,1)*xi(3,6))/k}$ deltaprimemodg(4,1):=0$ deltaprimemodg(4,2):=0$ deltaprimemodg(5,1):=0$ deltaprimemodg(5,2):=0$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=k$ deltaprimemodg(3,4):=0$ deltaprimemodg(3,5):=0$ deltaprimemodg(3,6):=(b(1,1)*xi(3,6))/k$ det(AUTOM):=(b(6,6)**3*b(5,5)**2*b(1,1)**2)/k$ DELTAPRIMEMODADG:= [0 0 0 0 0 0 ] [ ] [0 0 0 0 0 0 ] [ ] [ b(1,1)*xi(3,6) ] [0 0 0 0 0 ----------------] [ k ] [ ] [0 0 0 0 k 0 ] [ ] [0 0 0 0 0 0 ] [ ] [0 k 0 0 0 0 ] We are reduced in that case to : $ (0,0;0,0;0,1;0,0,epsilon) with epsilon=xi(3,6)=0,1.$