Book a Demo!
CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutPoliciesSign UpSign In
Download

Environment to perform calculations of equivariant vector bundles on homogeneous varieties

1842 views
License: GPL3
ubuntu2204
Tweet
Share
Share
Kernel: SageMath 9.8
%run '/home/user/Equivariant_Vector_Bundles_On_Homogeneous_Varieties__0-0-1/src/Initialize.ipynb'

k = 3
n = 4
N = 2*n+1
X = Orthogonal_Grassmannian(k,N)
print( 'X:' , X )
print( '(n='+str(n)+')' )
print()

G = X.Parent_Group()
fw = X.Basis('fw')

print( 'Initialise suitable objects:' )
print( 3*' ' , '- Repesentations')
rV = G.rmV(fw[1])
rS = G.rmV(fw[n])
print( 3*' ' , '- Tautological bundles')
cO = X.calO()
cU = X.calU()
cQ = dict({})
# Consturct Wedge(p) of cQ
for p in [ 0 .. 2*n-2 ] :
    Dict = dict({})
    Dict.update({ -p+i-1 : cU.Symmetric_Power(p-i)*rV.exterior_power(i) for i in [ 0 .. p ] })
    Dict.update({      0 : 'Cokernel' })
    cQ.update({ tuple(p*[1]) : X.Complex(Dict).SemiSimplification(0) })

print( 3*' ' , '- Spinor bundles' )
cS = dict({})
cS.update({ 0 : X.calS() })
cS.update({ y : ( cU.Dual().Symmetric_Power(y)*cS[0] ).Extend_Equivariantly_By( cU.Dual().Symmetric_Power(y-1)*cS[0] ) for y in [ 1 .. n-2 ] })
print( 3*' ' , '- Mysterious object' )
cM = cQ[(1,1,)].Extend_Equivariantly_By( cO )
print()

print('Initialise Lefschetz collection:')
TC = X.Tautological_Collection( Parts=[1,2] )
MC = cM.Maximal_Exceptional_Orbit( Test_Numerically=True )
SC = X.Spinor_Collection()
LC = TC.Subcollection( Columns=[0,1] )+MC+TC.Subcollection( Columns=[2] )+SC
print( table([ [ x , Obj ] for x , Obj in enumerate( LC.Starting_Block() ) ]) )
print()
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 9-dimensional ambient vector space. (n=4) Initialise suitable objects: - Repesentations - Tautological bundles - Spinor bundles - Mysterious object Initialise Lefschetz collection: 0 VB(0) 1 VB(Lambda[1]) 2 Equivariant extension of VB(Lambda[1] - Lambda[3] + 2*Lambda[4]) + VB(Lambda[2]) + VB(-Lambda[3] + 2*Lambda[4]) by VB(0) 3 VB(Lambda[2]) 4 VB(Lambda[4]) 5 Equivariant extension of VB(Lambda[1] + Lambda[4]) by VB(Lambda[4]) 6 Equivariant extension of VB(2*Lambda[1] + Lambda[4]) by VB(Lambda[1] + Lambda[4]) 
# Consturct Schur(2,1) of cQ
Dict = dict({})
Dict.update({ -4 : X.calU( fw[1]+fw[2] ).Dual()                                                                        ,
              -3 : ( cU.Symmetric_Power(2) + cU.Exterior_Power(2) ) * ( rV                                           ) ,
              -2 : ( cU                                           ) * ( rV.symmetric_power(2) + rV.exterior_power(2) ) ,
              -1 : ( cO                                           ) * ( G.rmV( fw[1]+fw[2] )                         ) ,
               0 : 'Cokernel'
            })
cQ.update({ (2,1) : X.Complex(Dict).SemiSimplification(0) })

# Consturct Schur(2,2) of cQ
Dict = dict({})
Dict.update({ -5 : X.calU( 2*fw[2] ).Dual()                                     ,
              -4 : ( X.calU( fw[1]+fw[2] ).Dual() ) * ( rV                    ) ,
              -3 : ( cU.Symmetric_Power(2)        ) * ( rV.exterior_power(2)  ) + \
                   ( cU.Exterior_Power(2)         ) * ( rV.symmetric_power(2) ) ,
              -2 : ( cU                           ) * ( G.rmV( fw[1]+fw[2] )  ) ,
              -1 : ( cO                           ) * ( G.rmV( 2*fw[2] )      ) ,
               0 : 'Cokernel'
            })
cQ.update({ (2,2) : X.Complex(Dict).SemiSimplification(0) })

# Consturct Schur(3,1) of cQ
Dict = dict({})
Dict.update({ -5 : X.calU( fw[1]+fw[3] ).Dual()                                                        ,
              -4 : ( X.calU( fw[1]+fw[2] ).Dual() ) * ( rV                                           ) ,
              -3 : ( cU.Symmetric_Power(2)        ) * ( rV.symmetric_power(2)                        ) + \
                   ( cU.Exterior_Power(2)         ) * ( rV.exterior_power(2)                         ) ,
              -2 : ( cU                           ) * ( G.rmV( fw[1]+fw[2] ) + rV.symmetric_power(3) ) ,
              -1 : ( cO                           ) * ( G.rmV( 2*fw[1]+fw[2] )                       ) ,
               0 : 'Cokernel'
            })
cQ.update({ (3,1) : X.Complex(Dict).SemiSimplification(0) })



print( 'On the sequence 0 -> cE -> ... -> cE(l) -> 0 whenever cE is from the starting block of the tautological collection' )
print()

for t , cE in  [ ( 1 , cO ) ,
                 #( 2 , cU.Dual() ) ,
                 #( 3 , cU.Dual().Wedge(2) )
               ] :
               #enumerate( TC.Starting_Block() , start=1 ) :
    print( 150*'-' )
    l = len(cE.Maximal_Exceptional_Orbit(Test_Numerically=True))
    print( bcolors.OKBLUE+'cE = '+str(cE)+' with l = '+str(l)+bcolors.ENDC )
    print( bcolors.OKBLUE+'(t='+str(t)+')'+bcolors.ENDC )
    print()

    print( 'Initialise the left side.' )
    cA = dict({})
    Objects = dict({})
    Objects.update({ -1*i-1 : cU.Exterior_Power(i)(1) * rV.symmetric_power(4-t-i) for i in [ 0 .. 4-t ] })
    Objects.update({      0 : 'Cokernel' })
    cA.update({ 1 : X.Complex( Objects ) })
    Objects = dict({})
    Objects.update({ -2 : cQ[(1,)].Symmetric_Power(4-t)(1)                        ,
                     -1 : cQ[(1,)].Symmetric_Power(4-t)(2) * rV.exterior_power(5)
                   })
    Objects.update({      0 : 'Cokernel' })
    cA.update({ 2 : X.Complex( Objects ) })

    print( 'Initialise the right side.' )
    cB = dict({})
    if t == 1 :
        Objects = dict({})
        Objects.update({ 0 : 'Kernel'                                                     ,
                         1 : cQ[(1,)].Dual().Symmetric_Power(2)(4) * rV.exterior_power(1) ,
                         2 : cQ[(1,)].Dual().Symmetric_Power(1)(4) * rV.exterior_power(2) ,
                         3 : cQ[(1,)].Dual().Symmetric_Power(0)(4) * rV.exterior_power(3) ,
                         4 : cO(5)
                       })
        cB.update({ -1 : X.Complex( Objects ) })
        Objects = dict({})
        Objects.update({ 0 : 'Kernel'                                                     ,
                         1 : cQ[(1,)].Dual().Symmetric_Power(2)(3) * rV.exterior_power(5) ,
                         2 : cQ[(1,)].Dual().Symmetric_Power(3)(4)
                       })
        cB.update({ -2 : X.Complex( Objects ) })
    elif t == 2 :
        Objects = dict({})
        Objects.update({ 0 : 'Kernel'                                                                                                      ,
                         1 : ( cQ[(1,)].Dual().Symmetric_Power(2) * cQ[(1,)].Dual() )(4) * ( rV                                          ) ,
                         2 : ( cQ[(1,1,)].Dual()                                    )(4) * ( rV.exterior_power(2)                        )  + \
                             ( cQ[(1,)].Dual().Symmetric_Power(2)                   )(4) * ( rV.symmetric_power(2)                       ) ,
                         3 : ( cQ[(1,)].Dual()                                      )(4) * ( G.rmV( fw[1]+fw[2] ) + rV.exterior_power(3) ) ,
                         4 : ( cO                                                   )(4) * ( G.rmV( fw[1]+fw[3] )                        ) ,
                         5 : cU.Dual()(5)
                       })
        cB.update({ -1 : X.Complex( Objects ) })
    elif t == 3 :
        Objects = dict({})
        Objects.update({ 0 : 'Kernel'                                                                                   ,
                         1 : ( cQ[(3,1,)].Dual() + cQ[(2,2,)].Dual() )(4) * ( rV ) ,
                         2 : ( cQ[(1,)].Dual().Symmetric_Power(3)    )(4) * ( rV.exterior_power(2)                    ) + \
                             ( cQ[(2,1,)].Dual()                     )(4) * ( rV.symmetric_power(2)                   ) ,
                         3 : ( cQ[(1,1,)].Dual()                     )(4) * ( G.rmV( fw[1]+fw[2] )                    ) + \
                             ( cQ[(1,)].Dual().Symmetric_Power(2)    )(4) * ( rV.symmetric_power(3)                   ) ,
                         4 : ( cQ[(1,)].Dual()                       )(4) * ( G.rmV( fw[1]+fw[3] ) + G.rmV( 2*fw[2] ) ) ,
                         5 : ( cO                                    )(4) * ( G.rmV( fw[2]+fw[3] )                    ) ,
                         6 : cU.Dual().Wedge(2)(5)
                       })
        cB.update({ -1 : X.Complex( Objects ) })

    print()

    for Label , Stack in [ ( 'A' , cA ) , ( 'B' , cB ) ] :
        print( 'Construction of '+Label+'´s:' )
        for Counter , Complex in Stack.items() :
            print( 'Counter='+str(Counter)+':' )
            print( 'Complex:')
            print( Complex )
            if Complex.Is_Numerically_Exact() :
                print( 'Is complex numerically exact? '+bcolors.OKGREEN+'Yes.'+bcolors.ENDC )
            else :
                print( 'Is complex numerically exact? '+bcolors.FAIL+'No.'+bcolors.ENDC )
                #print( '-> Cohomology:' )
                #for Degree , NC in Complex.Numerical_Cohomology().items() :
                #    print( 3*' ' , Degree , NC )
            #print()
            #print( table( [ [ 'Smd' , 'Smd.Dual()(5)' ] ] + [ [ Smd , Smd.Dual()(5) ] for Smd in Complex.SemiSimplification(0) ] , header_row=True ) )
            print()

    for BCounter , BCplx in cB.items() :
        for ACounter , ACplx in cA.items() :

            print('Compute EXTs from B['+str(BCounter)+'][*] to A['+str(ACounter)+'][*]:')

            print( 'Objects:' )
            for Label , Cplx in [ ( 'B' , BCplx ) , ( 'A' , ACplx ) ] :
                if   Label == 'A' : Counter = ACounter
                elif Label == 'B' : Counter = BCounter
                print( table([ [ Label+'['+str(Counter)+']['+str(Degree)+']' , '=' , Component ] for Degree , Component in Cplx ]) )
                print()

            Header = [ [ '' ] + [ 'A['+str(ACounter)+']['+str(ADegree)+']' for ADegree , AComponent in ACplx ] ]
            Body = []
            for BDegree , BComponent in BCplx :
                Row = [ 'B['+str(BCounter)+']['+str(BDegree)+']' ]
                for ADegree , AComponent in ACplx :
                    if isinstance( BComponent , str ) or isinstance( AComponent , str ) : Result = '...'
                    else                                                                :
                        if BComponent.Euler_Sum(AComponent) == G.rmV(0) : Result = '{}'
                        else                                            : Result = BComponent.EXT(AComponent)
                    Row += [ Result ]
                Body += [ Row ]
            show( table( Header + Body , header_column=True , header_row=True ) )
            print()
On the sequence 0 -> cE -> ... -> cE(l) -> 0 whenever cE is from the starting block of the tautological collection ------------------------------------------------------------------------------------------------------------------------------------------------------ cE = VB(0) with l = 5 (t=1) Initialise the left side. Initialise the right side. Construction of A´s: Counter=1: Complex: ... --d_-6--> 0 --d_-5--> VB(0) --d_-4--> B4(1,0,0,0) * VB(Lambda[1]) --d_-3--> ( B4(0,0,0,0) + B4(2,0,0,0) ) * VB(Lambda[2]) --d_-2--> ( B4(1,0,0,0) + B4(3,0,0,0) ) * VB(Lambda[3]) --d_-1--> Cokernel --d_0--> 0 --d_1--> ... Is complex numerically exact? Yes. Counter=2: Complex: ... --d_-4--> 0 --d_-3--> VB(3*Lambda[1] + Lambda[3]) + VB(2*Lambda[1] + 2*Lambda[4]) + VB(Lambda[1] - Lambda[3] + 4*Lambda[4]) + VB(Lambda[1] + Lambda[3]) + VB(-2*Lambda[3] + 6*Lambda[4]) + VB(2*Lambda[4]) --d_-2--> B4(0,0,0,2) * VB(3*Lambda[1] + 2*Lambda[3]) + B4(0,0,0,2) * VB(2*Lambda[1] + Lambda[3] + 2*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[1] + 4*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[1] + 2*Lambda[3]) + B4(0,0,0,2) * VB(-Lambda[3] + 6*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[3] + 2*Lambda[4]) --d_-1--> Cokernel --d_0--> 0 --d_1--> ... Is complex numerically exact? Yes. Construction of B´s: Counter=-1: Complex: ... --d_-2--> 0 --d_-1--> Kernel --d_0--> B4(1,0,0,0) * VB(2*Lambda[2] + 2*Lambda[3]) + B4(1,0,0,0) * VB(Lambda[2] + 2*Lambda[3] + 2*Lambda[4]) + B4(1,0,0,0) * VB(2*Lambda[3] + 4*Lambda[4]) + B4(1,0,0,0) * VB(4*Lambda[3]) --d_1--> B4(0,1,0,0) * VB(Lambda[2] + 3*Lambda[3]) + B4(0,1,0,0) * VB(3*Lambda[3] + 2*Lambda[4]) --d_2--> B4(0,0,1,0) * VB(4*Lambda[3]) --d_3--> VB(5*Lambda[3]) --d_4--> 0 --d_5--> ... Is complex numerically exact? Yes. Counter=-2: Complex: ... --d_-2--> 0 --d_-1--> Kernel --d_0--> B4(0,0,0,2) * VB(2*Lambda[2] + Lambda[3]) + B4(0,0,0,2) * VB(Lambda[2] + Lambda[3] + 2*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[3] + 4*Lambda[4]) + B4(0,0,0,2) * VB(3*Lambda[3]) --d_1--> VB(3*Lambda[2] + Lambda[3]) + VB(2*Lambda[2] + Lambda[3] + 2*Lambda[4]) + VB(Lambda[2] + Lambda[3] + 4*Lambda[4]) + VB(Lambda[2] + 3*Lambda[3]) + VB(Lambda[3] + 6*Lambda[4]) + VB(3*Lambda[3] + 2*Lambda[4]) --d_2--> 0 --d_3--> ... Is complex numerically exact? Yes. Compute EXTs from B[-1][*] to A[1][*]: Objects: B[-1][0] = Kernel B[-1][1] = B4(1,0,0,0) * VB(2*Lambda[2] + 2*Lambda[3]) + B4(1,0,0,0) * VB(Lambda[2] + 2*Lambda[3] + 2*Lambda[4]) + B4(1,0,0,0) * VB(2*Lambda[3] + 4*Lambda[4]) + B4(1,0,0,0) * VB(4*Lambda[3]) B[-1][2] = B4(0,1,0,0) * VB(Lambda[2] + 3*Lambda[3]) + B4(0,1,0,0) * VB(3*Lambda[3] + 2*Lambda[4]) B[-1][3] = B4(0,0,1,0) * VB(4*Lambda[3]) B[-1][4] = VB(5*Lambda[3]) A[1][-4] = VB(0) A[1][-3] = B4(1,0,0,0) * VB(Lambda[1]) A[1][-2] = ( B4(0,0,0,0) + B4(2,0,0,0) ) * VB(Lambda[2]) A[1][-1] = ( B4(1,0,0,0) + B4(3,0,0,0) ) * VB(Lambda[3]) A[1][0] = Cokernel
Compute EXTs from B[-1][*] to A[2][*]: Objects: B[-1][0] = Kernel B[-1][1] = B4(1,0,0,0) * VB(2*Lambda[2] + 2*Lambda[3]) + B4(1,0,0,0) * VB(Lambda[2] + 2*Lambda[3] + 2*Lambda[4]) + B4(1,0,0,0) * VB(2*Lambda[3] + 4*Lambda[4]) + B4(1,0,0,0) * VB(4*Lambda[3]) B[-1][2] = B4(0,1,0,0) * VB(Lambda[2] + 3*Lambda[3]) + B4(0,1,0,0) * VB(3*Lambda[3] + 2*Lambda[4]) B[-1][3] = B4(0,0,1,0) * VB(4*Lambda[3]) B[-1][4] = VB(5*Lambda[3]) A[2][-2] = VB(3*Lambda[1] + Lambda[3]) + VB(2*Lambda[1] + 2*Lambda[4]) + VB(Lambda[1] - Lambda[3] + 4*Lambda[4]) + VB(Lambda[1] + Lambda[3]) + VB(-2*Lambda[3] + 6*Lambda[4]) + VB(2*Lambda[4]) A[2][-1] = B4(0,0,0,2) * VB(3*Lambda[1] + 2*Lambda[3]) + B4(0,0,0,2) * VB(2*Lambda[1] + Lambda[3] + 2*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[1] + 4*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[1] + 2*Lambda[3]) + B4(0,0,0,2) * VB(-Lambda[3] + 6*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[3] + 2*Lambda[4]) A[2][0] = Cokernel
Compute EXTs from B[-2][*] to A[1][*]: Objects: B[-2][0] = Kernel B[-2][1] = B4(0,0,0,2) * VB(2*Lambda[2] + Lambda[3]) + B4(0,0,0,2) * VB(Lambda[2] + Lambda[3] + 2*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[3] + 4*Lambda[4]) + B4(0,0,0,2) * VB(3*Lambda[3]) B[-2][2] = VB(3*Lambda[2] + Lambda[3]) + VB(2*Lambda[2] + Lambda[3] + 2*Lambda[4]) + VB(Lambda[2] + Lambda[3] + 4*Lambda[4]) + VB(Lambda[2] + 3*Lambda[3]) + VB(Lambda[3] + 6*Lambda[4]) + VB(3*Lambda[3] + 2*Lambda[4]) A[1][-4] = VB(0) A[1][-3] = B4(1,0,0,0) * VB(Lambda[1]) A[1][-2] = ( B4(0,0,0,0) + B4(2,0,0,0) ) * VB(Lambda[2]) A[1][-1] = ( B4(1,0,0,0) + B4(3,0,0,0) ) * VB(Lambda[3]) A[1][0] = Cokernel
Compute EXTs from B[-2][*] to A[2][*]: Objects: B[-2][0] = Kernel B[-2][1] = B4(0,0,0,2) * VB(2*Lambda[2] + Lambda[3]) + B4(0,0,0,2) * VB(Lambda[2] + Lambda[3] + 2*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[3] + 4*Lambda[4]) + B4(0,0,0,2) * VB(3*Lambda[3]) B[-2][2] = VB(3*Lambda[2] + Lambda[3]) + VB(2*Lambda[2] + Lambda[3] + 2*Lambda[4]) + VB(Lambda[2] + Lambda[3] + 4*Lambda[4]) + VB(Lambda[2] + 3*Lambda[3]) + VB(Lambda[3] + 6*Lambda[4]) + VB(3*Lambda[3] + 2*Lambda[4]) A[2][-2] = VB(3*Lambda[1] + Lambda[3]) + VB(2*Lambda[1] + 2*Lambda[4]) + VB(Lambda[1] - Lambda[3] + 4*Lambda[4]) + VB(Lambda[1] + Lambda[3]) + VB(-2*Lambda[3] + 6*Lambda[4]) + VB(2*Lambda[4]) A[2][-1] = B4(0,0,0,2) * VB(3*Lambda[1] + 2*Lambda[3]) + B4(0,0,0,2) * VB(2*Lambda[1] + Lambda[3] + 2*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[1] + 4*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[1] + 2*Lambda[3]) + B4(0,0,0,2) * VB(-Lambda[3] + 6*Lambda[4]) + B4(0,0,0,2) * VB(Lambda[3] + 2*Lambda[4]) A[2][0] = Cokernel 


print( 'On the sequence 0 -> cE -> ... -> cE(l) -> 0 whenever cE is from the starting block of the spinor collection' )
print()

for y in  [ 0 , 1 , 2
          ] :
    print( 150*'-' )
    cE = cS[y]
    l = len(cE.Maximal_Exceptional_Orbit(Test_Numerically=True))
    print( bcolors.OKBLUE+'cE = '+str(cE)+' with l = '+str(l)+bcolors.ENDC )
    print( bcolors.OKBLUE+'(t='+str(t)+')'+bcolors.ENDC )
    print()

    print( 'Initialise the left side.' )
    cA = dict({})
    if y == 0 :
        Objects = dict({})
        Objects.update({ -1 : cE         ,
                          0 : 'Cokernel'   #cE.Dual()(1)
                       })
        cA.update({ 1 : X.Complex( Objects ) })
    elif y == 1 :
        Objects = dict({})
        Objects.update({ -2 : cE           ,
                         -1 : cO(1) * rS   ,
                          0 : 'Cokernel'     #cE.Dual()(2)
                       })
        cA.update({ 1 : X.Complex( Objects ) })

    elif y == 2 :
        Objects = dict({})
        Objects.update({ -3 : cE                         ,
                         -2 : cU.Dual()(1)          * rS ,
                         -1 : cU.Dual().Wedge(2)(1) * rS ,
                          0 : 'Cokernel'                   #cE.Dual()(3)
                       })
        cA.update({ 1 : X.Complex( Objects ) })

    print( 'Initialise the right side.' )
    cB = dict({})
    if y == 0 :
        Objects = dict({})
        Objects.update({ 0 : 'Kernel'                      ,
                         1 : cO(4) * cE(1).Cohomology()[0] ,
                         2 : cE(5)
                       })
        cB.update({ -1 : X.Complex( Objects ) })
    elif y == 1 :
        Objects = dict({})
        Objects.update({ 0 : 'Kernel'                      ,
                         1 : cO(4) * cE(1).Cohomology()[0] ,
                         2 : cE(5)
                       })
        cB.update({ -1 : X.Complex( Objects ) })
    elif y == 2 :
        Objects = dict({})
        Objects.update({ 0 : 'Kernel'                      ,
                         1 : cO(1) * cE(1).Cohomology()[0] ,
                         2 : cE(2)
                       })
        cB.update({ -1 : X.Complex( Objects ) })

    print()

    for Label , Stack in [ ( 'A' , cA ) , ( 'B' , cB ) ] :
        print( 'Construction of '+Label+'´s:' )
        for Counter , Complex in Stack.items() :
            print( 'Counter='+str(Counter)+':' )
            print( 'Complex:')
            print( Complex )
            if Complex.Is_Numerically_Exact() :
                print( 'Is complex numerically exact? '+bcolors.OKGREEN+'Yes.'+bcolors.ENDC )
            else :
                print( 'Is complex numerically exact? '+bcolors.FAIL+'No.'+bcolors.ENDC )
                #print( '-> Cohomology:' )
                #for Degree , NC in Complex.Numerical_Cohomology().items() :
                #    print( 3*' ' , Degree , NC )
            #print()
            #print( table( [ [ 'Smd' , 'Smd.Dual()(5)' ] ] + [ [ Smd , Smd.Dual()(5) ] for Smd in Complex.SemiSimplification(0) ] , header_row=True ) )
            print()

    for BCounter , BCplx in cB.items() :
        for ACounter , ACplx in cA.items() :

            print('Compute EXTs from B['+str(BCounter)+'][*] to A['+str(ACounter)+'][*]:')

            print( 'Objects:' )
            for Label , Cplx in [ ( 'B' , BCplx ) , ( 'A' , ACplx ) ] :
                if   Label == 'A' : Counter = ACounter
                elif Label == 'B' : Counter = BCounter
                print( table([ [ Label+'['+str(Counter)+']['+str(Degree)+']' , '=' , Component ] for Degree , Component in Cplx ]) )
                print()

            Header = [ [ '' ] + [ 'A['+str(ACounter)+']['+str(ADegree)+']' for ADegree , AComponent in ACplx ] ]
            Body = []
            for BDegree , BComponent in BCplx :
                Row = [ 'B['+str(BCounter)+']['+str(BDegree)+']' ]
                for ADegree , AComponent in ACplx :
                    if isinstance( BComponent , str ) or isinstance( AComponent , str ) : Result = '...'
                    else                                                                :
                        if BComponent.Euler_Sum(AComponent) == G.rmV(0) : Result = '{}'
                        else                                            : Result = BComponent.EXT(AComponent)
                    Row += [ Result ]
                Body += [ Row ]
            show( table( Header + Body , header_column=True , header_row=True ) )
            print()
On the sequence 0 -> cE -> ... -> cE(l) -> 0 whenever cE is from the starting block of the spinor collection ------------------------------------------------------------------------------------------------------------------------------------------------------ cE = VB(Lambda[4]) with l = 5 (t=1) Initialise the left side. Initialise the right side. Construction of A´s: Counter=1: Complex: ... --d_-3--> 0 --d_-2--> VB(Lambda[4]) --d_-1--> Cokernel --d_0--> 0 --d_1--> ... Is complex numerically exact? Yes. Construction of B´s: Counter=-1: Complex: ... --d_-2--> 0 --d_-1--> Kernel --d_0--> B4(0,0,1,1) * VB(4*Lambda[3]) --d_1--> VB(5*Lambda[3] + Lambda[4]) --d_2--> 0 --d_3--> ... Is complex numerically exact? Yes. Compute EXTs from B[-1][*] to A[1][*]: Objects: B[-1][0] = Kernel B[-1][1] = B4(0,0,1,1) * VB(4*Lambda[3]) B[-1][2] = VB(5*Lambda[3] + Lambda[4]) A[1][-1] = VB(Lambda[4]) A[1][0] = Cokernel
------------------------------------------------------------------------------------------------------------------------------------------------------ cE = Equivariant extension of VB(Lambda[1] + Lambda[4]) by VB(Lambda[4]) with l = 5 (t=1) Initialise the left side. Initialise the right side.
--------------------------------------------------------------------------- ValueError Traceback (most recent call last) Cell In [29], line 53 48 Objects = dict({}) 49 Objects.update({ Integer(0) : 'Kernel' , 50 Integer(1) : cO(Integer(4)) * cE(Integer(1)).Cohomology()[Integer(0)] , 51 Integer(2) : cE(Integer(5)) 52 }) ---> 53 cB.update({ -Integer(1) : X.Complex( Objects ) }) 54 elif y == Integer(2) : 55 Objects = dict({})
File /tmp/ipykernel_30382/1270618750.py:261, in Irreducible_Homogeneous_Variety.Complex(self, Objects) 259 def Complex ( self , Objects ) -> "Complex_Over_Irreducible_Homogeneous_Variety" : 260 """Returns a complex of equivariant vector bundles over ``self``.""" --> 261 return Complex_Of_Coherent_Sheaves( Base_Space=self , Objects=Objects )
File /tmp/ipykernel_30382/1415231974.py:47, in Complex_Of_Coherent_Sheaves.__init__(self, Base_Space, Objects) 45 if not Current_Degree in self._Numerical_Cohomology.keys() : 46 Numerical_Kernel_In_Previous_Degree , Numerical_Image_In_Previous_Degree = self.Decompose_Differential_Numerically(Current_Degree-Integer(1)) ---> 47 Numerical_Kernel_In_Current_Degree , Numerical_Image_In_Current_Degree = self.Decompose_Differential_Numerically(Current_Degree) 48 if Numerical_Image_In_Previous_Degree in Numerical_Kernel_In_Current_Degree : 49 Numerical_Cohomology_Group = Numerical_Kernel_In_Current_Degree - Numerical_Image_In_Previous_Degree
File /tmp/ipykernel_30382/1415231974.py:90, in Complex_Of_Coherent_Sheaves.Decompose_Differential_Numerically(self, Degree) 87 Numerical_Image_In_Current_Degree = Domain - Numerical_Kernel_In_Current_Degree 89 elif Current_Object == 'Kernel' : ---> 90 Numerical_Kernel_In_Subsequent_Degree , Numerical_Image_In_Subsequent_Degree = self.Decompose_Differential_Numerically( Degree+Integer(1) ) 91 Numerical_Kernel_In_Current_Degree = self.Zero_Object() 92 Numerical_Image_In_Current_Degree = Numerical_Kernel_In_Subsequent_Degree
File /tmp/ipykernel_30382/1415231974.py:83, in Complex_Of_Coherent_Sheaves.Decompose_Differential_Numerically(self, Degree) 81 else : 82 Domain = self.Zero_Object() ---> 83 for Summand in Current_Object.Irreducible_Components() : 84 Domain += Summand 85 Numerical_Kernel_In_Subsequent_Degree , Numerical_Image_In_Subsequent_Degree = self.Decompose_Differential_Numerically( Degree+Integer(1) )
File /tmp/ipykernel_30382/3374629093.py:329, in Direct_Sum_Of_Equivariant_Vector_Bundles.Irreducible_Components(self) 327 for Counter in (ellipsis_range( Integer(1) ,Ellipsis, Multiplicity )) : 328 rk_2 = self.Base_Space().calU( Weight ).Rank() --> 329 Dict[rk_2].remove(Weight)
ValueError: list.remove(x): x not in list 


cE = cS[1]

K = X.Zero_Vector_Bundle()
for Smd in [ cO(4) * G.rmV(       fw[3]+fw[4] ) ,
             cO(4) * G.rmV( fw[1]+fw[3]+fw[4] )
           ] :
    print( 'Smd:' , Smd )
    K += Smd.SemiSimplification()
K += cE(5).SemiSimplification() * -1

for Smd in K :
    print(Smd)
#print( K )
Smd: B4(0,0,1,1) * VB(4*Lambda[3]) Smd: B4(1,0,1,1) * VB(4*Lambda[3]) ( B4(0,0,1,1) + B4(1,0,1,1) ) * VB(4*Lambda[3]) -B4(0,0,0,0) * VB(Lambda[1] + 5*Lambda[3] + Lambda[4]) -B4(0,0,0,0) * VB(5*Lambda[3] + Lambda[4])