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Environment to perform calculations of equivariant vector bundles on homogeneous varieties

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License: GPL3
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Kernel: SageMath 9.8
%run './Initialize.ipynb'
from Equivariant_Vector_Bundles_On_Homogeneous_Varieties.Base_Space.Orthogonal_Grassmannian import Orthogonal_Grassmannian

X = Orthogonal_Grassmannian( 2 , 9 )
n = X.Cartan_Degree()
print( X )
print()

#assert X.Is_Adjoint() , \
#       'The homogeneous variety X is not adjoint.'

r = (X.Dimension()-1)/2
print( 'We have '+str(X.Dimension())+' = dim X = 2*r+1 for r='+str(r)+'.' )
print()

#CM = X.Cartan_Matrix( From='ambt' , To='fw' )

i = 3
print( 'Decompose H^0( G/P , wedge^'+str(i)+' T_{G/P} ):' )
Restiction = 'reg'
if i <= r :
    for p in [ 0 .. floor(i/2) ] :
        print( 'Kostant´s space for ('+str(i-2*p)+','+str(p)+'):' )
        for Weight in X.Kostant_Space(i-2*p,p,Restiction) :
            print( 3*' ' , Weight )

    for p in [ 0 .. floor(i/2-1/2) ] :
        print( 'Kostant´s space for ('+str(i-2*p)+','+str(p)+'):' )
        for Weight in X.Kostant_Space(i-2*p-1,p+1,Restiction) :
            print( 3*' ' , Weight )

else :
    for p in [ 0 .. r-ceil(i/2) ] :
        print( 'Kostant´s space for ('+str(i+1+2*p)+','+str(-p-1)+'):' )
        for Weight in X.Kostant_Space(i+1+2*p,-p-1,Restiction) :
            print( 3*' ' , Weight )
    for p in [ 0 .. r-ceil(i/2-1/2) ] :
        print( 'Kostant´s space for ('+str(i+2*p)+','+str(-p-2)+'):' )
        for Weight in X.Kostant_Space(i+2*p,-p-2,Restiction) :
            print( 3*' ' , Weight )
Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 9-dimensional ambient vector space. We have 11 = dim X = 2*r+1 for r=5. Decompose H^0( G/P , wedge^3 T_{G/P} ): Kostant´s space for (3,0): Kostant´s space for (1,1): Lambda[1] + Lambda[3] Kostant´s space for (3,0): 2*Lambda[1] + 2*Lambda[4] 2*Lambda[3] Kostant´s space for (1,1): 2*Lambda[2] 








k = 3
for n in [ 5 .. 7 ] :
    N = 2*n+1
    X = Orthogonal_Grassmannian(k,N)
    G = X.Parent_Group()

    print( 'X:' , X )
    print( '(n='+str(n)+')' )
    d = X.Dimension()
    print( 3*' ' , 'Dimension: '+str(d) )
    lMax = X.K0().rank()
    print( 3*' ' , 'Rank of K0(X) (max. collection length): '+str(lMax) )
    wMax = X.Fano_Index()
    print( 3*' ' , 'Fano index (max. orbit length): '+str(wMax) )
    print()

    fw = X.Basis('fw')

    print( 3*' ' , 'Consecutive Lefschetz collection.' )
    LC1 = X.My_Collection( Modus='Con' )
    l1 = len(LC1)
    print( 3*' ' , 'Starting block:' )
    w1_0 = len( LC1.Starting_Block() )
    s = ceil( math.log10( w1_0 ) )
    for i , cE in enumerate ( LC1.Starting_Block() , start=1 ) :
        print( 6*' ' , (s-floor(math.log10(i))-1)*' '+str(i) , cE )
    print()

    print( 3*' ' , 'Alternating Lefschetz collection.' )
    LC2 = X.My_Collection( Modus='Alt' )
    l2 = len(LC2)
    print( 6*' ' , 'Starting block:' )
    w2_0 = len( LC2.Starting_Block() )
    s = ceil( math.log10( w2_0 ) )
    for i , cE in enumerate ( LC2.Starting_Block() , start=1 ) :
        print( 6*' ' , (s-floor(math.log10(i))-1)*' '+str(i) , cE )
    print()
    print( 3*' ' , 'Gap.')
    print( 6*' ' , '       lMax-l = '+str(lMax-l1) )
    print( 6*' ' , '(lMax-l)/wMax = '+str( (lMax-l1)/wMax  ) )
    print()

    print( 3*' ' , 'Try to fill the gap.' )
    Start_With = 0
    print( 3*' ' , 'Start later than candidate #'+str(Start_With)+'.')
    print()
    Counter = 0
    for Node in [ 4 .. n ] :
        Universe = [ E for E in ( X.calU( fw[Node] ) * G.rmV( fw[n] ) ).Irreducible_Components() ]
        for Size in [ 1 .. len(Universe) ] :
            for Indices in Subsets( range(len(Universe)) , Size , submultiset=True ) :
                # Construct candidate.
                Counter += 1
                if Start_With < Counter :
                    Candidate = X.Zero_Vector_Bundle()
                    for Index in Indices :
                        Candidate += Universe[Index]
                    # Check if candidate is admissible.
                    Candidate_Is_Admissible = True
                    if not Candidate.Is_Exceptional( Test_Numerically=True ) :
                        Candidate_Is_Admissible = False
                    if Candidate_Is_Admissible :
                        print( 6*' ' , 'Test for candidate #'+str(Counter)+':' )
                        print( 6*' ' , '(ss = '+str(Candidate)+')' )
                        for i , LC in enumerate([ LC1 , LC2 ] , start=1 ) :
                            Stack = [ ( xPos , Admissible_Columns ) for xPos , Admissible_Columns in LC.Test_For_Extension ( New_Object=Candidate , Test_Numerically=True , Test_If_Self_Is_Exceptional=False ) if 0 < len(Admissible_Columns) ]
                            if 0 < len(Stack) :
                                print( 9*' ' , 'Candidate can be embedded to LC'+str(i)+' in the row ...' )
                                for xPos , Admissible_Columns in Stack :
                                    print( 12*' ' , '... '+str(xPos)+' to the columns '+str(Admissible_Columns) )
                                print( )
                        print()

    print()
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 11-dimensional ambient vector space. (n=5) Dimension: 18 Rank of K0(X) (max. collection length): 80 Fano index (max. orbit length): 7 Consecutive Lefschetz collection. Starting block: 1 VB(0) 2 VB(Lambda[1]) 3 VB(Lambda[2]) 4 VB(2*Lambda[1]) 5 VB(Lambda[1] + Lambda[2]) 6 VB(2*Lambda[2]) 7 VB(2*Lambda[1] + Lambda[2]) 8 VB(Lambda[5]) 9 Equivariant extension of VB(Lambda[1] + Lambda[5]) by VB(Lambda[5]) 10 Equivariant extension of VB(2*Lambda[1] + Lambda[5]) by VB(Lambda[1] + Lambda[5]) 11 Equivariant extension of VB(3*Lambda[1] + Lambda[5]) by VB(2*Lambda[1] + Lambda[5]) Alternating Lefschetz collection. Starting block: 1 VB(0) 2 VB(Lambda[5]) 3 VB(Lambda[1]) 4 VB(Lambda[2]) 5 Equivariant extension of VB(Lambda[1] + Lambda[5]) by VB(Lambda[5]) 6 VB(2*Lambda[1]) 7 VB(Lambda[1] + Lambda[2]) 8 VB(2*Lambda[2]) 9 Equivariant extension of VB(2*Lambda[1] + Lambda[5]) by VB(Lambda[1] + Lambda[5]) 10 VB(2*Lambda[1] + Lambda[2]) 11 Equivariant extension of VB(3*Lambda[1] + Lambda[5]) by VB(2*Lambda[1] + Lambda[5]) Gap. lMax-l = 7 (lMax-l)/wMax = 1 Try to fill the gap. Start later than candidate #0. Test for candidate #5: (ss = VB(Lambda[4] + Lambda[5])) Test for candidate #6: (ss = VB(Lambda[3] + Lambda[5])) Test for candidate #7: (ss = VB(-Lambda[3] + Lambda[4] + Lambda[5])) Test for candidate #8: (ss = VB(Lambda[5])) Test for candidate #19: (ss = VB(Lambda[2] + Lambda[5]) + VB(Lambda[3] + Lambda[5])) Test for candidate #30: (ss = VB(Lambda[1] + Lambda[5]) + VB(Lambda[5])) Test for candidate #64: (ss = VB(Lambda[2] + Lambda[5]) + VB(Lambda[1] + Lambda[5]) + VB(Lambda[3] + Lambda[5])) Test for candidate #66: (ss = VB(Lambda[2] + Lambda[5]) + VB(Lambda[1] + Lambda[5]) + VB(Lambda[5])) Test for candidate #67: (ss = VB(Lambda[2] + Lambda[5]) + VB(Lambda[4] + Lambda[5]) + VB(Lambda[3] + Lambda[5])) Test for candidate #88: (ss = VB(Lambda[1] + Lambda[5]) + VB(-Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[5])) Test for candidate #167: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[2] + Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[4] + Lambda[5]) + VB(Lambda[3] + Lambda[5])) Test for candidate #188: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[1] + Lambda[5]) + VB(-Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[5])) Test for candidate #247: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[2] + Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[1] + Lambda[5]) + VB(Lambda[4] + Lambda[5]) + VB(Lambda[3] + Lambda[5]) + VB(-Lambda[3] + Lambda[4] + Lambda[5])) Test for candidate #248: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[2] + Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[1] + Lambda[5]) + VB(Lambda[4] + Lambda[5]) + VB(Lambda[3] + Lambda[5]) + VB(Lambda[5])) Test for candidate #249: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[2] + Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[1] + Lambda[5]) + VB(Lambda[4] + Lambda[5]) + VB(-Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[5])) Test for candidate #250: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[2] + Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[1] + Lambda[5]) + VB(Lambda[3] + Lambda[5]) + VB(-Lambda[3] + Lambda[4] + Lambda[5]) + VB(Lambda[5])) Test for candidate #257: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4])) Test for candidate #258: (ss = VB(Lambda[2])) Test for candidate #260: (ss = VB(Lambda[1] - Lambda[3] + Lambda[4])) Test for candidate #261: (ss = VB(Lambda[1])) Test for candidate #262: (ss = VB(2*Lambda[5])) Test for candidate #263: (ss = VB(Lambda[4])) Test for candidate #264: (ss = VB(Lambda[3])) Test for candidate #265: (ss = VB(-Lambda[3] + 2*Lambda[5])) Test for candidate #266: (ss = VB(-Lambda[3] + Lambda[4])) Test for candidate #267: (ss = VB(0)) Test for candidate #273: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(2*Lambda[5])) Test for candidate #283: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(2*Lambda[5])) Test for candidate #285: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[3])) Test for candidate #293: (ss = VB(Lambda[2]) + VB(Lambda[4])) Test for candidate #303: (ss = VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + 2*Lambda[5])) Test for candidate #310: (ss = VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(-Lambda[3] + 2*Lambda[5])) Test for candidate #312: (ss = VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(0)) Test for candidate #317: (ss = VB(Lambda[1]) + VB(-Lambda[3] + Lambda[4])) Test for candidate #374: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(2*Lambda[5]) + VB(Lambda[4])) Test for candidate #415: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[1]) + VB(Lambda[3])) Test for candidate #420: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(2*Lambda[5]) + VB(Lambda[3])) Test for candidate #448: (ss = VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(0)) Test for candidate #495: (ss = VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + Lambda[4])) Candidate can be embedded to LC1 in the row ... ... 7 to the columns [0, 1, 2, 3, 4, 5, 6] Test for candidate #517: (ss = VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(0)) Test for candidate #620: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2]) + VB(2*Lambda[5]) + VB(Lambda[4])) Test for candidate #805: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[1]) + VB(2*Lambda[5]) + VB(Lambda[3])) Test for candidate #886: (ss = VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(0)) Test for candidate #956: (ss = VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + Lambda[4])) Test for candidate #2125: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(2*Lambda[5]) + VB(Lambda[4]) + VB(0)) Test for candidate #2480: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1]) + VB(Lambda[3]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + Lambda[4])) Test for candidate #3649: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(2*Lambda[5]) + VB(Lambda[4]) + VB(Lambda[3]) + VB(0)) Test for candidate #3719: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1]) + VB(2*Lambda[5]) + VB(Lambda[4]) + VB(Lambda[3]) + VB(-Lambda[3] + Lambda[4])) Test for candidate #3800: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(Lambda[4]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + Lambda[4]) + VB(0)) Test for candidate #3985: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(Lambda[1]) + VB(Lambda[3]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + Lambda[4]) + VB(0)) Test for candidate #4088: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1]) + VB(2*Lambda[5]) + VB(Lambda[4]) + VB(Lambda[3]) + VB(-Lambda[3] + Lambda[4])) Test for candidate #4110: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(Lambda[1]) + VB(2*Lambda[5]) + VB(Lambda[4]) + VB(Lambda[3]) + VB(0)) Candidate can be embedded to LC1 in the row ... ... 8 to the columns [0, 1, 2, 3, 4, 5, 6] ... 9 to the columns [0, 1, 2, 3, 4, 5, 6] ... 10 to the columns [0, 1, 2, 3, 4, 5, 6] ... 11 to the columns [0, 1, 2, 3, 4, 5, 6] Candidate can be embedded to LC2 in the row ... ... 10 to the columns [0, 1, 2, 3, 4, 5, 6] ... 11 to the columns [0, 1, 2, 3, 4, 5, 6] Test for candidate #4185: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(Lambda[1]) + VB(Lambda[4]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + Lambda[4]) + VB(0)) Test for candidate #4231: (ss = VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(Lambda[1]) + VB(Lambda[3]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + Lambda[4]) + VB(0)) Test for candidate #4288: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[2]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(2*Lambda[5]) + VB(Lambda[4]) + VB(Lambda[3]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(0)) Test for candidate #4312: (ss = VB(Lambda[2] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[1] - Lambda[3] + 2*Lambda[5]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + VB(Lambda[1]) + VB(2*Lambda[5]) + VB(Lambda[3]) + VB(-Lambda[3] + 2*Lambda[5]) + VB(-Lambda[3] + Lambda[4]) + VB(0)) 






k = 3
Rows = []
Header = 3*[ 'n' , 'n-'+str(k) , 'delta' , 'ceil(delta)' ]
Rows += [ Header ]
for m in [ 1 .. 20 ] :
    Row = []
    for r in [ 0 .. 2 ] :
        n = 3*m+r
        delta = 1/24*n^2 -1/8*n -1/8*n*(-1)^n +31/48 +11/16*(-1)^n
        Row += [ n , n-k , delta , ceil(delta) ]
    Rows += [ Row ]
show( table( Rows , frame=True , header_column=True ) )









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])
cE = B4(1,1,1,0) * VB(0) Rank = 9009 2*B4(0,0,0,0) * VB(2*Lambda[1] + 2*Lambda[2] - 4*Lambda[3] + 4*Lambda[4]) VB(3*Lambda[1] + 2*Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) VB(2*Lambda[1] + 3*Lambda[2] - 4*Lambda[3] + 2*Lambda[4]) VB(3*Lambda[1] + Lambda[2] - 3*Lambda[3] + 4*Lambda[4]) VB(Lambda[1] + 3*Lambda[2] - 5*Lambda[3] + 4*Lambda[4]) VB(2*Lambda[1] + Lambda[2] - 4*Lambda[3] + 6*Lambda[4]) VB(Lambda[1] + 2*Lambda[2] - 5*Lambda[3] + 6*Lambda[4]) 2*B4(0,0,0,0) * VB(2*Lambda[1] + 2*Lambda[2] - 2*Lambda[3] + 2*Lambda[4]) 2*B4(0,0,0,0) * VB(2*Lambda[1] + 2*Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) 2*B4(0,0,0,0) * VB(2*Lambda[1] + 2*Lambda[2] - 4*Lambda[3] + 2*Lambda[4]) VB(2*Lambda[1] + Lambda[2] - 2*Lambda[3] + 4*Lambda[4]) 3*B4(0,0,0,0) * VB(Lambda[1] + 2*Lambda[2] - 3*Lambda[3] + 4*Lambda[4]) 3*B4(0,0,0,0) * VB(2*Lambda[1] + Lambda[2] - 3*Lambda[3] + 4*Lambda[4]) 3*B4(0,0,0,0) * VB(Lambda[1] + 2*Lambda[2] - 4*Lambda[3] + 4*Lambda[4]) 3*B4(0,0,0,0) * VB(2*Lambda[1] + Lambda[2] - 4*Lambda[3] + 4*Lambda[4]) VB(Lambda[1] + 2*Lambda[2] - 5*Lambda[3] + 4*Lambda[4]) VB(3*Lambda[1] + Lambda[2] - 2*Lambda[3] + 2*Lambda[4]) 3*B4(0,0,0,0) * VB(Lambda[1] + 3*Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) 3*B4(0,0,0,0) * VB(3*Lambda[1] + Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) VB(Lambda[1] + 3*Lambda[2] - 4*Lambda[3] + 2*Lambda[4]) VB(Lambda[1] + Lambda[2] - 3*Lambda[3] + 6*Lambda[4]) 2*B4(0,0,0,0) * VB(Lambda[1] + Lambda[2] - 4*Lambda[3] + 6*Lambda[4]) VB(Lambda[1] + Lambda[2] - 5*Lambda[3] + 6*Lambda[4]) VB(3*Lambda[1] - 2*Lambda[3] + 4*Lambda[4]) 2*B4(0,0,0,0) * VB(3*Lambda[2] - 4*Lambda[3] + 4*Lambda[4]) 2*B4(0,0,0,0) * VB(3*Lambda[1] - 3*Lambda[3] + 4*Lambda[4]) VB(3*Lambda[2] - 5*Lambda[3] + 4*Lambda[4]) VB(Lambda[1] + 2*Lambda[2] - Lambda[3] + 2*Lambda[4]) VB(2*Lambda[1] + Lambda[2] - Lambda[3] + 2*Lambda[4]) 3*B4(0,0,0,0) * VB(Lambda[1] + 2*Lambda[2] - 2*Lambda[3] + 2*Lambda[4]) 9*B4(0,0,0,0) * VB(2*Lambda[1] + Lambda[2] - 2*Lambda[3] + 2*Lambda[4]) 9*B4(0,0,0,0) * VB(Lambda[1] + 2*Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) VB(4*Lambda[1] - 2*Lambda[3] + 2*Lambda[4]) 3*B4(0,0,0,0) * VB(2*Lambda[1] + Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) VB(4*Lambda[2] - 4*Lambda[3] + 2*Lambda[4]) VB(Lambda[1] + 2*Lambda[2] - 4*Lambda[3] + 2*Lambda[4]) VB(2*Lambda[1] + Lambda[2] - 4*Lambda[3] + 2*Lambda[4]) VB(2*Lambda[1] - 3*Lambda[3] + 6*Lambda[4]) VB(2*Lambda[1] + 3*Lambda[2] - 2*Lambda[3]) VB(2*Lambda[2] - 4*Lambda[3] + 6*Lambda[4]) VB(2*Lambda[1] - 4*Lambda[3] + 6*Lambda[4]) VB(2*Lambda[2] - 5*Lambda[3] + 6*Lambda[4]) VB(3*Lambda[1] + 2*Lambda[2] - 3*Lambda[3]) 2*B4(0,0,0,0) * VB(Lambda[1] + Lambda[2] - 2*Lambda[3] + 4*Lambda[4]) 8*B4(0,0,0,0) * VB(Lambda[1] + Lambda[2] - 3*Lambda[3] + 4*Lambda[4]) 2*B4(0,0,0,0) * VB(Lambda[1] + Lambda[2] - 4*Lambda[3] + 4*Lambda[4]) VB(4*Lambda[1] + Lambda[2] - 2*Lambda[3]) VB(Lambda[1] + 4*Lambda[2] - 3*Lambda[3]) 2*B4(0,0,0,0) * VB(3*Lambda[1] - Lambda[3] + 2*Lambda[4]) VB(2*Lambda[2] - 2*Lambda[3] + 4*Lambda[4]) 2*B4(0,0,0,0) * VB(3*Lambda[2] - 2*Lambda[3] + 2*Lambda[4]) 4*B4(0,0,0,0) * VB(2*Lambda[1] - 2*Lambda[3] + 4*Lambda[4]) 3*B4(0,0,0,0) * VB(2*Lambda[2] - 3*Lambda[3] + 4*Lambda[4]) 2*B4(0,0,0,0) * VB(3*Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) 2*B4(0,0,0,0) * VB(3*Lambda[1] - 2*Lambda[3] + 2*Lambda[4]) 3*B4(0,0,0,0) * VB(2*Lambda[1] - 3*Lambda[3] + 4*Lambda[4]) 4*B4(0,0,0,0) * VB(2*Lambda[2] - 4*Lambda[3] + 4*Lambda[4]) 2*B4(0,0,0,0) * VB(3*Lambda[2] - 4*Lambda[3] + 2*Lambda[4]) 2*B4(0,0,0,0) * VB(3*Lambda[1] - 3*Lambda[3] + 2*Lambda[4]) VB(2*Lambda[1] - 4*Lambda[3] + 4*Lambda[4]) 4*B4(0,0,0,0) * VB(2*Lambda[1] + 2*Lambda[2] - 2*Lambda[3]) VB(Lambda[1] + 3*Lambda[2] - Lambda[3]) 2*B4(0,0,0,0) * VB(3*Lambda[1] + Lambda[2] - Lambda[3]) 6*B4(0,0,0,0) * VB(Lambda[1] + Lambda[2] - Lambda[3] + 2*Lambda[4]) 8*B4(0,0,0,0) * VB(Lambda[1] + Lambda[2] - 2*Lambda[3] + 2*Lambda[4]) 6*B4(0,0,0,0) * VB(Lambda[1] + Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) 2*B4(0,0,0,0) * VB(Lambda[1] + 3*Lambda[2] - 3*Lambda[3]) VB(3*Lambda[1] + Lambda[2] - 3*Lambda[3]) VB(Lambda[2] - 3*Lambda[3] + 6*Lambda[4]) 2*B4(0,0,0,0) * VB(Lambda[1] - 3*Lambda[3] + 6*Lambda[4]) 2*B4(0,0,0,0) * VB(Lambda[2] - 4*Lambda[3] + 6*Lambda[4]) VB(Lambda[1] - 4*Lambda[3] + 6*Lambda[4]) VB(2*Lambda[1] + 2*Lambda[4]) VB(2*Lambda[2] - Lambda[3] + 2*Lambda[4]) 4*B4(0,0,0,0) * VB(2*Lambda[1] - Lambda[3] + 2*Lambda[4]) 6*B4(0,0,0,0) * VB(2*Lambda[2] - 2*Lambda[3] + 2*Lambda[4]) 6*B4(0,0,0,0) * VB(2*Lambda[1] - 2*Lambda[3] + 2*Lambda[4]) 4*B4(0,0,0,0) * VB(2*Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) VB(2*Lambda[1] - 3*Lambda[3] + 2*Lambda[4]) VB(2*Lambda[2] - 4*Lambda[3] + 2*Lambda[4]) 2*B4(0,0,0,0) * VB(2*Lambda[1] + Lambda[2]) VB(Lambda[1] - Lambda[3] + 4*Lambda[4]) 6*B4(0,0,0,0) * VB(Lambda[1] + 2*Lambda[2] - Lambda[3]) VB(4*Lambda[2] - 2*Lambda[3]) 4*B4(0,0,0,0) * VB(Lambda[2] - 2*Lambda[3] + 4*Lambda[4]) 5*B4(0,0,0,0) * VB(Lambda[1] - 2*Lambda[3] + 4*Lambda[4]) 6*B4(0,0,0,0) * VB(2*Lambda[1] + Lambda[2] - 2*Lambda[3]) 5*B4(0,0,0,0) * VB(Lambda[2] - 3*Lambda[3] + 4*Lambda[4]) 4*B4(0,0,0,0) * VB(Lambda[1] - 3*Lambda[3] + 4*Lambda[4]) 2*B4(0,0,0,0) * VB(Lambda[1] + 2*Lambda[2] - 3*Lambda[3]) VB(4*Lambda[1] - 2*Lambda[3]) VB(Lambda[2] - 4*Lambda[3] + 4*Lambda[4]) 3*B4(0,0,0,0) * VB(3*Lambda[2] - 2*Lambda[3]) 3*B4(0,0,0,0) * VB(3*Lambda[1] - Lambda[3]) VB(Lambda[2] + 2*Lambda[4]) VB(Lambda[1] + 2*Lambda[4]) 4*B4(0,0,0,0) * VB(Lambda[2] - Lambda[3] + 2*Lambda[4]) 7*B4(0,0,0,0) * VB(Lambda[1] - Lambda[3] + 2*Lambda[4]) 7*B4(0,0,0,0) * VB(Lambda[2] - 2*Lambda[3] + 2*Lambda[4]) 4*B4(0,0,0,0) * VB(Lambda[1] - 2*Lambda[3] + 2*Lambda[4]) VB(Lambda[2] - 3*Lambda[3] + 2*Lambda[4]) VB(Lambda[1] - 3*Lambda[3] + 2*Lambda[4]) VB(Lambda[1] + Lambda[2] + Lambda[3]) VB(Lambda[1] + Lambda[2]) 8*B4(0,0,0,0) * VB(Lambda[1] + Lambda[2] - Lambda[3]) VB(Lambda[1] + Lambda[2] - 2*Lambda[3]) VB(Lambda[1] + Lambda[2] - 3*Lambda[3]) VB(-3*Lambda[3] + 6*Lambda[4]) 2*B4(0,0,0,0) * VB(2*Lambda[2]) 3*B4(0,0,0,0) * VB(2*Lambda[1]) 2*B4(0,0,0,0) * VB(2*Lambda[2] - Lambda[3]) 2*B4(0,0,0,0) * VB(2*Lambda[1] - Lambda[3]) 3*B4(0,0,0,0) * VB(2*Lambda[2] - 2*Lambda[3]) 2*B4(0,0,0,0) * VB(2*Lambda[1] - 2*Lambda[3]) VB(-Lambda[3] + 4*Lambda[4]) 4*B4(0,0,0,0) * VB(-2*Lambda[3] + 4*Lambda[4]) VB(-3*Lambda[3] + 4*Lambda[4]) 2*B4(0,0,0,0) * VB(Lambda[1] + Lambda[3]) 2*B4(0,0,0,0) * VB(2*Lambda[4]) 3*B4(0,0,0,0) * VB(Lambda[2]) 3*B4(0,0,0,0) * VB(Lambda[1]) 4*B4(0,0,0,0) * VB(-Lambda[3] + 2*Lambda[4]) 3*B4(0,0,0,0) * VB(Lambda[2] - Lambda[3]) 3*B4(0,0,0,0) * VB(Lambda[1] - Lambda[3]) 2*B4(0,0,0,0) * VB(-2*Lambda[3] + 2*Lambda[4]) 2*B4(0,0,0,0) * VB(Lambda[2] - 2*Lambda[3]) 2*B4(0,0,0,0) * VB(0)
--------------------------------------------------------------------------- AssertionError Traceback (most recent call last) Cell In [147], line 11 7 print( 'Rank =' , cE_Rank ) 9 print() ---> 11 for IC in cE.Irreducible_Components() : 12 print( Integer(3)*' ' , IC )
File /tmp/ipykernel_29381/424321877.py:337, in Direct_Sum_Of_Equivariant_Vector_Bundles.Irreducible_Components(self) 334 yield Irreducible_Component 335 Summed_Ranks += Irreducible_Component.Rank() --> 337 assert Summed_Ranks == self.Rank() , 'The rank of ``self`` is '+str(self.Rank())+'. However, the sum over the ranks of all irreducible components is different, namely '+str(Summed_Ranks)+'.'
AssertionError: The rank of ``self`` is 9009. However, the sum over the ranks of all irreducible components is different, namely 9045.