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Environment to perform calculations of equivariant vector bundles on homogeneous varieties

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License: GPL3
Image: ubuntu2204
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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

for n in [ 2 .. 10 ] :
    print( 'n='+str(n)+':' )
    for k in [ 1 .. n ] :
        print( 3*' ' , 'k='+str(k)+':' )
        N = 2*n+1
        X = Orthogonal_Grassmannian(k,N)

        G = X.Parent_Group()
        fw = X.Basis('fw')
        rV = G.rmV(fw[1])
        cO = X.calO()
        cU = X.calU()
        cQ = X.Complex({ -2 : cU , -1 : cO * rV , 0 : 'Cokernel' }).SemiSimplification(0)
        cT = cU.Dual()*cQ - cU.Dual().Symmetric_Power(2)


        Results = []
        Highest_Degree = -infinity
        for q in [ 3 ] : #[ n .. cT.Rank() ] :
            cA = cT.Wedge(q)

            for t in [ 1 ] :
                cB = cO(t)

                EXTSpace = cA.EXT( cB )
                if 0 < max( list(EXTSpace.keys()) + [ -infinity ] ) and sum([ (-1)^Degree*Value for Degree , Value in EXTSpace.items() ]) != G.rmV(0) :
                    Results += [ ( q , t , EXTSpace ) ]
                    Highest_Degree = max([ Highest_Degree ] + list(EXTSpace.keys()) )

        if 0 < len(Results) :
            Output = [ [ '' ] + [ str(Degree) for Degree in [ 0 .. Highest_Degree ] ] ]
            for q , t , EXTSpace in Results :
                Values = []
                for CurrentDegree in [ 0 .. Highest_Degree ] :
                    if CurrentDegree in EXTSpace.keys() : Values += [ str(EXTSpace[CurrentDegree]) ]
                    else                                : Values += [ '' ]
                Output += [ [ 'EXT( cT.Wedge('+str(q)+') , cO('+str(t)+') )' ] + Values ]
            for Row in str(table( Output , header_row=True , header_column=True )).split('\n') :
                print( 6*' ' , Row )

        print()
n=2: k=1: k=2: n=3: k=1: | 0 1 2 +----------------------------+---+---+-----------+ EXT( cT.Wedge(3) , cO(1) ) | B3(0,0,0) k=2: | 0 1 2 +----------------------------+---+-----------+-------------+ EXT( cT.Wedge(3) , cO(1) ) | B3(0,0,0) 2*B3(0,0,0) k=3: n=4: k=1: k=2: | 0 1 2 +----------------------------+---+-------------+-----------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B4(0,0,0,0) 2*B4(0,0,0,0) + B4(1,0,0,0) k=3: | 0 1 2 +----------------------------+---+---------------------------+-----------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B4(0,0,0,0) + B4(1,0,0,0) B4(0,0,0,0) + 2*B4(1,0,0,0) k=4: n=5: k=1: k=2: | 0 1 2 +----------------------------+---+---------------+-----------------+ EXT( cT.Wedge(3) , cO(1) ) | B5(0,0,0,0,0) 2*B5(0,0,0,0,0) k=3: | 0 1 2 +----------------------------+---+---------------+---------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B5(1,0,0,0,0) 2*B5(1,0,0,0,0) + B5(0,1,0,0,0) k=4: | 0 1 2 +----------------------------+---------------+-----------------------------------------------+---------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B5(0,0,0,0,0) B5(0,0,0,0,0) + B5(1,0,0,0,0) + B5(0,1,0,0,0) B5(1,0,0,0,0) + 2*B5(0,1,0,0,0) k=5: n=6: k=1: k=2: | 0 1 2 +----------------------------+---+-----------------+-------------------+ EXT( cT.Wedge(3) , cO(1) ) | B6(0,0,0,0,0,0) 2*B6(0,0,0,0,0,0) k=3: | 0 1 2 +----------------------------+---+-----------------+-------------------+ EXT( cT.Wedge(3) , cO(1) ) | B6(1,0,0,0,0,0) 2*B6(1,0,0,0,0,0) k=4: | 0 1 2 +----------------------------+-----------------+-----------------------------------+-------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B6(0,0,0,0,0,0) B6(0,0,0,0,0,0) + B6(0,1,0,0,0,0) 2*B6(0,1,0,0,0,0) + B6(0,0,1,0,0,0) k=5: | 0 1 2 +----------------------------+-----------------+-----------------------------------------------------+-------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B6(1,0,0,0,0,0) B6(1,0,0,0,0,0) + B6(0,1,0,0,0,0) + B6(0,0,1,0,0,0) B6(0,1,0,0,0,0) + 2*B6(0,0,1,0,0,0) k=6: n=7: k=1: k=2: | 0 1 2 +----------------------------+---+-------------------+---------------------+ EXT( cT.Wedge(3) , cO(1) ) | B7(0,0,0,0,0,0,0) 2*B7(0,0,0,0,0,0,0) k=3: | 0 1 2 +----------------------------+---+-------------------+---------------------+ EXT( cT.Wedge(3) , cO(1) ) | B7(1,0,0,0,0,0,0) 2*B7(1,0,0,0,0,0,0) k=4: | 0 1 2 +----------------------------+-------------------+---------------------------------------+---------------------+ EXT( cT.Wedge(3) , cO(1) ) | B7(0,0,0,0,0,0,0) B7(0,0,0,0,0,0,0) + B7(0,1,0,0,0,0,0) 2*B7(0,1,0,0,0,0,0) k=5: | 0 1 2 +----------------------------+-------------------+---------------------------------------+-----------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B7(1,0,0,0,0,0,0) B7(1,0,0,0,0,0,0) + B7(0,0,1,0,0,0,0) 2*B7(0,0,1,0,0,0,0) + B7(0,0,0,1,0,0,0) k=6: | 0 1 2 +----------------------------+-------------------+-----------------------------------------------------------+-----------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B7(0,1,0,0,0,0,0) B7(0,1,0,0,0,0,0) + B7(0,0,1,0,0,0,0) + B7(0,0,0,1,0,0,0) B7(0,0,1,0,0,0,0) + 2*B7(0,0,0,1,0,0,0) k=7: n=8: k=1: k=2: | 0 1 2 +----------------------------+---+---------------------+-----------------------+ EXT( cT.Wedge(3) , cO(1) ) | B8(0,0,0,0,0,0,0,0) 2*B8(0,0,0,0,0,0,0,0) k=3: | 0 1 2 +----------------------------+---+---------------------+-----------------------+ EXT( cT.Wedge(3) , cO(1) ) | B8(1,0,0,0,0,0,0,0) 2*B8(1,0,0,0,0,0,0,0) k=4: | 0 1 2 +----------------------------+---------------------+-------------------------------------------+-----------------------+ EXT( cT.Wedge(3) , cO(1) ) | B8(0,0,0,0,0,0,0,0) B8(0,0,0,0,0,0,0,0) + B8(0,1,0,0,0,0,0,0) 2*B8(0,1,0,0,0,0,0,0) k=5: | 0 1 2 +----------------------------+---------------------+-------------------------------------------+-----------------------+ EXT( cT.Wedge(3) , cO(1) ) | B8(1,0,0,0,0,0,0,0) B8(1,0,0,0,0,0,0,0) + B8(0,0,1,0,0,0,0,0) 2*B8(0,0,1,0,0,0,0,0) k=6: | 0 1 2 +----------------------------+---------------------+-------------------------------------------+---------------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B8(0,1,0,0,0,0,0,0) B8(0,1,0,0,0,0,0,0) + B8(0,0,0,1,0,0,0,0) 2*B8(0,0,0,1,0,0,0,0) + B8(0,0,0,0,1,0,0,0) k=7: | 0 1 2 +----------------------------+---------------------+-----------------------------------------------------------------+---------------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B8(0,0,1,0,0,0,0,0) B8(0,0,1,0,0,0,0,0) + B8(0,0,0,1,0,0,0,0) + B8(0,0,0,0,1,0,0,0) B8(0,0,0,1,0,0,0,0) + 2*B8(0,0,0,0,1,0,0,0) k=8: n=9: k=1: k=2: | 0 1 2 +----------------------------+---+-----------------------+-------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B9(0,0,0,0,0,0,0,0,0) 2*B9(0,0,0,0,0,0,0,0,0) k=3: | 0 1 2 +----------------------------+---+-----------------------+-------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B9(1,0,0,0,0,0,0,0,0) 2*B9(1,0,0,0,0,0,0,0,0) k=4: | 0 1 2 +----------------------------+-----------------------+-----------------------------------------------+-------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B9(0,0,0,0,0,0,0,0,0) B9(0,0,0,0,0,0,0,0,0) + B9(0,1,0,0,0,0,0,0,0) 2*B9(0,1,0,0,0,0,0,0,0) k=5: | 0 1 2 +----------------------------+-----------------------+-----------------------------------------------+-------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B9(1,0,0,0,0,0,0,0,0) B9(1,0,0,0,0,0,0,0,0) + B9(0,0,1,0,0,0,0,0,0) 2*B9(0,0,1,0,0,0,0,0,0) k=6: | 0 1 2 +----------------------------+-----------------------+-----------------------------------------------+-------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B9(0,1,0,0,0,0,0,0,0) B9(0,1,0,0,0,0,0,0,0) + B9(0,0,0,1,0,0,0,0,0) 2*B9(0,0,0,1,0,0,0,0,0) k=7: | 0 1 2 +----------------------------+-----------------------+-----------------------------------------------+-------------------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B9(0,0,1,0,0,0,0,0,0) B9(0,0,1,0,0,0,0,0,0) + B9(0,0,0,0,1,0,0,0,0) 2*B9(0,0,0,0,1,0,0,0,0) + B9(0,0,0,0,0,1,0,0,0) k=8: | 0 1 2 +----------------------------+-----------------------+-----------------------------------------------------------------------+-------------------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B9(0,0,0,1,0,0,0,0,0) B9(0,0,0,1,0,0,0,0,0) + B9(0,0,0,0,1,0,0,0,0) + B9(0,0,0,0,0,1,0,0,0) B9(0,0,0,0,1,0,0,0,0) + 2*B9(0,0,0,0,0,1,0,0,0) k=9: n=10: k=1: k=2: | 0 1 2 +----------------------------+---+--------------------------+----------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B10(0,0,0,0,0,0,0,0,0,0) 2*B10(0,0,0,0,0,0,0,0,0,0) k=3: | 0 1 2 +----------------------------+---+--------------------------+----------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B10(1,0,0,0,0,0,0,0,0,0) 2*B10(1,0,0,0,0,0,0,0,0,0) k=4: | 0 1 2 +----------------------------+--------------------------+-----------------------------------------------------+----------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B10(0,0,0,0,0,0,0,0,0,0) B10(0,0,0,0,0,0,0,0,0,0) + B10(0,1,0,0,0,0,0,0,0,0) 2*B10(0,1,0,0,0,0,0,0,0,0) k=5: | 0 1 2 +----------------------------+--------------------------+-----------------------------------------------------+----------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B10(1,0,0,0,0,0,0,0,0,0) B10(1,0,0,0,0,0,0,0,0,0) + B10(0,0,1,0,0,0,0,0,0,0) 2*B10(0,0,1,0,0,0,0,0,0,0) k=6: | 0 1 2 +----------------------------+--------------------------+-----------------------------------------------------+----------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B10(0,1,0,0,0,0,0,0,0,0) B10(0,1,0,0,0,0,0,0,0,0) + B10(0,0,0,1,0,0,0,0,0,0) 2*B10(0,0,0,1,0,0,0,0,0,0) k=7: | 0 1 2 +----------------------------+--------------------------+-----------------------------------------------------+----------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B10(0,0,1,0,0,0,0,0,0,0) B10(0,0,1,0,0,0,0,0,0,0) + B10(0,0,0,0,1,0,0,0,0,0) 2*B10(0,0,0,0,1,0,0,0,0,0) k=8: | 0 1 2 +----------------------------+--------------------------+-----------------------------------------------------+-------------------------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B10(0,0,0,1,0,0,0,0,0,0) B10(0,0,0,1,0,0,0,0,0,0) + B10(0,0,0,0,0,1,0,0,0,0) 2*B10(0,0,0,0,0,1,0,0,0,0) + B10(0,0,0,0,0,0,1,0,0,0) k=9: | 0 1 2 +----------------------------+--------------------------+--------------------------------------------------------------------------------+-------------------------------------------------------+ EXT( cT.Wedge(3) , cO(1) ) | B10(0,0,0,0,1,0,0,0,0,0) B10(0,0,0,0,1,0,0,0,0,0) + B10(0,0,0,0,0,1,0,0,0,0) + B10(0,0,0,0,0,0,1,0,0,0) B10(0,0,0,0,0,1,0,0,0,0) + 2*B10(0,0,0,0,0,0,1,0,0,0) k=10: 




class bcolors:
    HEADER = '\033[95m'
    OKBLUE = '\033[94m'
    OKCYAN = '\033[96m'
    OKGREEN = '\033[92m'
    WARNING = '\033[93m'
    FAIL = '\033[91m'
    ENDC = '\033[0m'
    BOLD = '\033[1m'
    UNDERLINE = '\033[4m'

k = 3

for n in [ 4 .. 15 ] :
    N = 2*n+1
    X = Orthogonal_Grassmannian(k,N)
    print( 'X:' , X )
    print( '(n='+str(n)+')' )
    G = X.Parent_Group()
    fw = X.Basis('fw')

    Store = { (1,2) : set({}) , (1,3) : set({}) , (2,3) : set({}) }
    for Counter1 , cE1 in enumerate( X.Tautological_Collection( Parts=[1] ).Starting_Block() , start=1 ) :
        for Counter2 , cE2 in enumerate( X.Tautological_Collection( Parts=[2] ).Starting_Block() , start=1 ) :
            Twists = [ 1 .. 2*n-4 ]
            for t in Twists :
                mu1 = cE1(t).Highest_Weight().to_ambient()
                mu2 = cE2.Highest_Weight().to_ambient()
                Bounds = { i : ( -mu1[3-i]+mu2[2] , -mu1[2]+mu2[i-1] )  for i in [ 1 , 2 , 3 ] }
                for Smd in cE1(t).Dual()*cE2 :
                    if Smd.Is_Irreducible() : Summands = [ ( Smd , G.rmV(1) ) ]
                    else                    : Summands = Smd.Summands()
                    for VB , WCR in Summands :
                        Highest_Weight = VB.Highest_Weight().to_ambient()
                        for i , j in Store.keys() :
                            Store[(i,j)].add( Highest_Weight[i-1]-Highest_Weight[j-1] )
    print( sorted(Store[(1,2)]) , ' has upper bound '+str(2*n-6) )
    print( sorted(Store[(1,3)]) , ' has upper bound '+str(floor(5/2*n-15/2)) )
    print( sorted(Store[(2,3)]) , ' has upper bound '+str(ceil(3/2*n-9/2)) )
    print()
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 9-dimensional ambient vector space. (n=4) [] has upper bound 2 [] has upper bound 2 [] has upper bound 2 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 11-dimensional ambient vector space. (n=5) [0, 1, 2, 3, 4] has upper bound 4 [1, 2, 3, 4, 5] has upper bound 5 [0, 1, 2, 3] has upper bound 3 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 13-dimensional ambient vector space. (n=6) [0, 1, 2, 3, 4, 5, 6] has upper bound 6 [1, 2, 3, 4, 5, 6, 7] has upper bound 7 [0, 1, 2, 3, 4, 5] has upper bound 5 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 15-dimensional ambient vector space. (n=7) [0, 1, 2, 3, 4, 5, 6, 7, 8] has upper bound 8 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] has upper bound 10 [0, 1, 2, 3, 4, 5, 6] has upper bound 6 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 17-dimensional ambient vector space. (n=8) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] has upper bound 10 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] has upper bound 12 [0, 1, 2, 3, 4, 5, 6, 7, 8] has upper bound 8 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 19-dimensional ambient vector space. (n=9) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] has upper bound 12 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] has upper bound 15 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] has upper bound 9 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 21-dimensional ambient vector space. (n=10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] has upper bound 14 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] has upper bound 17 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] has upper bound 11 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 23-dimensional ambient vector space. (n=11) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] has upper bound 16 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] has upper bound 20 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] has upper bound 12 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 25-dimensional ambient vector space. (n=12) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] has upper bound 18 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] has upper bound 22 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] has upper bound 14 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 27-dimensional ambient vector space. (n=13) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] has upper bound 20 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] has upper bound 25 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] has upper bound 15 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 29-dimensional ambient vector space. (n=14) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] has upper bound 22 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] has upper bound 27 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] has upper bound 17 X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 31-dimensional ambient vector space. (n=15) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] has upper bound 24 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] has upper bound 30 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] has upper bound 18 


class bcolors:
    HEADER = '\033[95m'
    OKBLUE = '\033[94m'
    OKCYAN = '\033[96m'
    OKGREEN = '\033[92m'
    WARNING = '\033[93m'
    FAIL = '\033[91m'
    ENDC = '\033[0m'
    BOLD = '\033[1m'
    UNDERLINE = '\033[4m'

k = 4
for n in [ k .. k+10 ] :
    N = 2*n+1
    X = Orthogonal_Grassmannian(k,N)
    print( 'n:' , n )
    print( 'X:' , X )
    wMax = X.Fano_Index()
    print( 'Fano index = '+bcolors.OKBLUE+str( wMax )+bcolors.ENDC )
    lMax = X.K0().rank()
    print( 'rk K_0(X) = '+bcolors.OKBLUE+str( lMax )+bcolors.ENDC )

    fw = X.Basis('fw')

    # Tautological collection and spinor collection
    print( 3*' ' , 'Initialise the tautological collection and the spinor collection, namely LC=TC+SC. ...' )

    LC = X.Tautological_Collection() + X.Spinor_Collection()
    LC_SupportPartition = tuple([ sum([ 1 for BlockEntry in Block if BlockEntry == True ])for BlockCounter , Block in LC.Support_Pattern().items() ])

    print( 3*' ' , 'LC has support partition '+bcolors.OKBLUE+str(LC_SupportPartition)+bcolors.ENDC+' and therefore consists of '+bcolors.OKBLUE+str(len(LC))+bcolors.ENDC+' objects.' )

    print()

    # Search for extensions of the spinor collection which can be merged to LC
    Universe = []
    for Factor1 in [ X.calU(d*fw[1]) for d in [ 0 .. n-k+1 ] ] :
        for Factor2 in [ X.calU(fw[n]) , X.calU(fw[1]+fw[n]) , X.calU(fw[2]+fw[n]) ] :
            for Summand in ( Factor1 * Factor2 ).Irreducible_Components() :
                if not Summand in Universe :
                    Universe += [ Summand ]

    print( 3*' ' , 'Test for objects which extend the spinor collection. ...' )

    SC2 = X.Lefschetz_Collection( Starting_Block=[] , Support_Pattern='Trivial' )
    ObjectCounter = 0
    for l in [ 2 , 3 ] :
        for Keys in Subsets( range(len(Universe)) , l , submultiset=True ) :
            ObjectCounter += 1
            Object = X.Zero_Vector_Bundle()
            for SummandPointer in Keys :
                Object += Universe[SummandPointer]
            if k == n : Object = Object(-1)
            Orbit = Object.Maximal_Numerically_Exceptional_Orbit()
            Orbit_Length = len( Orbit )
            if   0 < Orbit_Length :
                #hw1 = Object.Highest_Weight()
                #hw2 = tuple([ hw1.to_ambient().coefficient(Node) for Node in range(k) ])
                print( 6*' ' , '- ' , 'The object consisting of summands '+bcolors.OKBLUE+', '.join([ str(IrreducibleSummand) for IrreducibleSummand in Object.Irreducible_Components() ])+bcolors.ENDC+' '+ \
                                      'has orbit length ' , end=''
                     )
                if   Orbit_Length < wMax :
                    print( bcolors.OKCYAN+str(Orbit_Length)+bcolors.ENDC+'.' )
                elif Orbit_Length == wMax :
                    print( bcolors.OKGREEN+str(Orbit_Length)+bcolors.ENDC+'.' )
                print( 6*' ' , '  ' , 'Test if the object can be attached to LC: ' )
                Is_Object_Suitable = False
                for Column , Possible_Rows in LC.Test_For_Numerically_Exceptional_Proper_Extension( New_Object=Object , \
                                                                                                    Relevant_Columns=set([ len(X.Tautological_Collection().Starting_Block()) .. LC.Width()]) , \
                                                                                                    Test_If_Self_Is_Exceptional=False
                                                                                                  ) :
                    print( 6*' ' , '  ' , 3*' ' , 'Column='+bcolors.OKBLUE+str(Column)+bcolors.ENDC+' -> Admissible rows: '+bcolors.OKBLUE+str(Possible_Rows)+bcolors.ENDC+'.' )
                    if len(Possible_Rows) == wMax : Is_Object_Suitable = True
                if Is_Object_Suitable == True : SC2 += Orbit

    print()

    print( 3*' ' , 'The extension of the spinor collection consits of the following objects:' )
    if len(SC2) == 0 :
        print( 6*' ' , bcolors.FAIL+'Empty'+bcolors.ENDC+'.' )
    else :
        for Counter , Object in enumerate( SC2.Starting_Block() , start=1 ) :
            print( 6*' ' , Counter , Object )

        print()

        print( 3*' ' , 'It has the following non-admissible right-orthogonal relations:' )
        GramMatrix = []

        for x1 , Object1 in enumerate( SC2.Starting_Block() , start=1 ) :
            for x2 , Object2 in enumerate( SC2.Starting_Block() , start=1 ) :
                for y in [ 0 .. wMax-1 ] :
                    if y == 0 :
                        if x2 < x1 :
                            if 0 < len( Object1.EXT( Object2 ) ) :
                                print( 6*' ' , 'E_'+str(x1)+' is '+bcolors.FAIL+'NOT'+bcolors.ENDC+' right-orthogonal to '+'E_'+str(x2)+'.' )

                            elif x1 <= x2 :
                                pass

                    else :
                        if 0 < len( Object1(y).EXT( Object2 ) ) :
                            print( 6*' ' , 'E_'+str(x1)+'('+str(y)+') is '+bcolors.FAIL+'NOT'+bcolors.ENDC+' right-orthogonal to '+'E_'+str(x2)+'.' )

    print()
n: 4 X: Orthogonal grassmannian variety of 4-dimensional isotropic linear subspaces in a 9-dimensional ambient vector space. Fano index = 8 rk K_0(X) = 16 Initialise the tautological collection and the spinor collection, namely LC=TC+SC. ... LC has support partition (1, 1, 1, 1, 1, 1, 1, 1) and therefore consists of 8 objects. Test for objects which extend the spinor collection. ... - The object consisting of summands VB(0), VB(Lambda[1]) has orbit length 8. Test if the object can be attached to LC: Column=1 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7]. - The object consisting of summands VB(0), VB(Lambda[1]), VB(Lambda[2]) has orbit length 8. Test if the object can be attached to LC: Column=1 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7]. - The object consisting of summands VB(0), VB(Lambda[1]), VB(2*Lambda[1]) has orbit length 2. Test if the object can be attached to LC: Column=1 -> Admissible rows: [0, 1]. The extension of the spinor collection consits of the following objects: 1 VB(0) + VB(Lambda[1]) 2 VB(0) + VB(Lambda[1]) + VB(Lambda[2]) It has the following non-admissible right-orthogonal relations: E_1(2) is NOT right-orthogonal to E_2. E_2 is NOT right-orthogonal to E_1. E_2(6) is NOT right-orthogonal to E_1. E_2(2) is NOT right-orthogonal to E_2. E_2(6) is NOT right-orthogonal to E_2. n: 5 X: Orthogonal grassmannian variety of 4-dimensional isotropic linear subspaces in a 11-dimensional ambient vector space. Fano index = 6 rk K_0(X) = 80 Initialise the tautological collection and the spinor collection, namely LC=TC+SC. ... LC has support partition (6, 6, 6, 6, 6, 6) and therefore consists of 36 objects. Test for objects which extend the spinor collection. ... - The object consisting of summands VB(Lambda[5]), VB(Lambda[1] + Lambda[5]) has orbit length 6. Test if the object can be attached to LC: Column=4 -> Admissible rows: []. Column=5 -> Admissible rows: []. Column=6 -> Admissible rows: []. - The object consisting of summands VB(Lambda[1] + Lambda[5]), VB(2*Lambda[1] + Lambda[5]) has orbit length 6. Test if the object can be attached to LC: Column=4 -> Admissible rows: []. Column=5 -> Admissible rows: []. Column=6 -> Admissible rows: [0, 1, 2, 3, 4, 5]. - The object consisting of summands VB(2*Lambda[1] + Lambda[5]), VB(3*Lambda[1] + Lambda[5]) has orbit length 6. Test if the object can be attached to LC: Column=4 -> Admissible rows: []. Column=5 -> Admissible rows: []. Column=6 -> Admissible rows: []. - The object consisting of summands VB(Lambda[5]), VB(Lambda[1] + Lambda[5]), VB(Lambda[2] + Lambda[5]) has orbit length 6. Test if the object can be attached to LC: Column=4 -> Admissible rows: []. Column=5 -> Admissible rows: []. Column=6 -> Admissible rows: [0, 1, 2, 3, 4, 5]. The extension of the spinor collection consits of the following objects: 1 VB(Lambda[1] + Lambda[5]) + VB(2*Lambda[1] + Lambda[5]) 2 VB(Lambda[5]) + VB(Lambda[1] + Lambda[5]) + VB(Lambda[2] + Lambda[5]) It has the following non-admissible right-orthogonal relations: E_2 is NOT right-orthogonal to E_1. n: 6 X: Orthogonal grassmannian variety of 4-dimensional isotropic linear subspaces in a 13-dimensional ambient vector space. Fano index = 8 rk K_0(X) = 240 Initialise the tautological collection and the spinor collection, namely LC=TC+SC. ... LC has support partition (17, 17, 17, 17, 17, 17, 17, 17) and therefore consists of 136 objects. Test for objects which extend the spinor collection. ... - The object consisting of summands VB(Lambda[6]), VB(Lambda[1] + Lambda[6]) has orbit length 8. Test if the object can be attached to LC: Column=16 -> Admissible rows: []. Column=17 -> Admissible rows: []. Column=14 -> Admissible rows: []. Column=15 -> Admissible rows: []. - The object consisting of summands VB(Lambda[1] + Lambda[6]), VB(2*Lambda[1] + Lambda[6]) has orbit length 8. Test if the object can be attached to LC: Column=16 -> Admissible rows: []. Column=17 -> Admissible rows: []. Column=14 -> Admissible rows: []. Column=15 -> Admissible rows: []. - The object consisting of summands VB(2*Lambda[1] + Lambda[6]), VB(3*Lambda[1] + Lambda[6]) has orbit length 8. Test if the object can be attached to LC: Column=16 -> Admissible rows: []. Column=17 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7]. Column=14 -> Admissible rows: []. Column=15 -> Admissible rows: []. - The object consisting of summands VB(3*Lambda[1] + Lambda[6]), VB(4*Lambda[1] + Lambda[6]) has orbit length 8. Test if the object can be attached to LC: Column=16 -> Admissible rows: []. Column=17 -> Admissible rows: []. Column=14 -> Admissible rows: []. Column=15 -> Admissible rows: []. - The object consisting of summands VB(Lambda[6]), VB(Lambda[1] + Lambda[6]), VB(Lambda[2] + Lambda[6]) has orbit length 8. Test if the object can be attached to LC: Column=16 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7]. Column=17 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7]. Column=14 -> Admissible rows: []. Column=15 -> Admissible rows: []. The extension of the spinor collection consits of the following objects: 1 VB(2*Lambda[1] + Lambda[6]) + VB(3*Lambda[1] + Lambda[6]) 2 VB(Lambda[6]) + VB(Lambda[1] + Lambda[6]) + VB(Lambda[2] + Lambda[6]) It has the following non-admissible right-orthogonal relations: E_2 is NOT right-orthogonal to E_1. n: 7 X: Orthogonal grassmannian variety of 4-dimensional isotropic linear subspaces in a 15-dimensional ambient vector space. Fano index = 10 rk K_0(X) = 560 Initialise the tautological collection and the spinor collection, namely LC=TC+SC. ... LC has support partition (30, 30, 30, 30, 30, 30, 30, 30, 30, 30) and therefore consists of 300 objects. Test for objects which extend the spinor collection. ... - The object consisting of summands VB(Lambda[7]), VB(Lambda[1] + Lambda[7]) has orbit length 10. Test if the object can be attached to LC: Column=26 -> Admissible rows: []. Column=27 -> Admissible rows: []. Column=28 -> Admissible rows: []. Column=29 -> Admissible rows: []. Column=30 -> Admissible rows: []. - The object consisting of summands VB(Lambda[1] + Lambda[7]), VB(2*Lambda[1] + Lambda[7]) has orbit length 10. Test if the object can be attached to LC: Column=26 -> Admissible rows: []. Column=27 -> Admissible rows: []. Column=28 -> Admissible rows: []. Column=29 -> Admissible rows: []. Column=30 -> Admissible rows: []. - The object consisting of summands VB(2*Lambda[1] + Lambda[7]), VB(3*Lambda[1] + Lambda[7]) has orbit length 10. Test if the object can be attached to LC: Column=26 -> Admissible rows: []. Column=27 -> Admissible rows: []. Column=28 -> Admissible rows: []. Column=29 -> Admissible rows: []. Column=30 -> Admissible rows: []. - The object consisting of summands VB(3*Lambda[1] + Lambda[7]), VB(4*Lambda[1] + Lambda[7]) has orbit length 10. Test if the object can be attached to LC: Column=26 -> Admissible rows: []. Column=27 -> Admissible rows: []. Column=28 -> Admissible rows: []. Column=29 -> Admissible rows: []. Column=30 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]. - The object consisting of summands VB(4*Lambda[1] + Lambda[7]), VB(5*Lambda[1] + Lambda[7]) has orbit length 10. Test if the object can be attached to LC: Column=26 -> Admissible rows: []. Column=27 -> Admissible rows: []. Column=28 -> Admissible rows: []. Column=29 -> Admissible rows: []. Column=30 -> Admissible rows: []. - The object consisting of summands VB(Lambda[7]), VB(Lambda[1] + Lambda[7]), VB(Lambda[2] + Lambda[7]) has orbit length 10. Test if the object can be attached to LC: Column=26 -> Admissible rows: []. Column=27 -> Admissible rows: []. Column=28 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]. Column=29 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]. Column=30 -> Admissible rows: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9].