Object_cM1.ipynb

Kernel: SageMath 10.3

%run '../Initialize.ipynb'
from Equivariant_Vector_Bundles_On_Homogeneous_Varieties.Base_Space.Orthogonal_Grassmannian import Orthogonal_Grassmannian

k = 3

for n in [ 4 .. 10 ] :
    N = 2*n+1
    X = Orthogonal_Grassmannian(k,N)
    G = X.Parent_Group()
    fw = X.Basis('fw')

    print( 'X: '+str(X) )
    print( 3*' ' , '(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()

    if n == 4 : cR = X.calU( 2*fw[4] )(-1)
    else      : cR = X.calU(   fw[4] )(-1)

    cM2 = X.calU().Dual()*cR.Wedge(n-3) + cR.Wedge(n-3) + cR.Wedge(n-4)
    cM1 = cM2*-1
    for Smd in ( X.calS()*G.rmV( fw[n] ) ).Irreducible_Components() :
        cM1 += Smd

    #cQ = dict({})
    #for p in [ 0 .. n-1 ] :
    #    Stock = { -p-1+i : X.calU().Symmetric_Power(p-i) * G.rmV( fw[1] ).exterior_power(i)
    #              for i in [ 0 .. p ]
    #            }
    #    Stock.update({ 0 : 'Cokernel' })
    #    cQ.update({ p : X.Complex(Stock).SemiSimplification(0) })
    #
    #cM2 = X.Zero_Vector_Bundle()
    #for q in [ 0 .. floor((n-1)/2) ] :
    #    cM2 += cQ[2*q]
    #print( 3*' ' , 'cM2 has semi-simplification:' )
    #for Smd in cM2 :
    #      print( 6*' ' , bcolors.OKBLUE+str(Smd)+bcolors.ENDC )
    #print()
    #
    #cM1 = cM2*-1
    #for Smd in ( X.calS()*G.rmV( fw[n] ) ).Irreducible_Components() :
    #    cM1 += Smd
    #print( 3*' ' , 'cM1 has semi-simplification:' )
    #for Smd in cM1 :
    #      print( 6*' ' , bcolors.OKBLUE+str(Smd)+bcolors.ENDC )
    #print()

    Is_cM1_exceptional = cM1.Is_Exceptional( Test_Numerically=True )
    if Is_cM1_exceptional :
        print( 3*' ' , 'cM1 is '+bcolors.OKGREEN+'exceptional'+bcolors.ENDC+'.' )
        print()

        for i , LC in enumerate([ X.My_Collection( Modus='Con' ) , X.My_Collection( Modus='Alt' ) ] , start=1 ) :
            print( 3*' ' , 'Can candidate be embedded in the Lefschetz collection LC'+str(i)+'?' )
            for xPos , Admissible_Columns in LC.Test_For_Extension ( New_Object=cM1 , Test_Numerically=True , Test_If_Self_Is_Exceptional=False ) :
                if 0 < len(Admissible_Columns) : print( 6*' ' , bcolors.OKGREEN+'Yes'+bcolors.ENDC+' after row '+str(xPos)+' to the columns '+bcolors.OKBLUE+str(Admissible_Columns)+bcolors.ENDC+'.' )
                else                           : print( 6*' ' , bcolors.FAIL+'Not'+bcolors.ENDC+' after row '+str(xPos)+'.' )
            print()

    else :
        print( 3*' ' , 'cM1 is '+bcolors.FAIL+'not exceptional'+bcolors.ENDC+'.' )
        print()

X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 9-dimensional ambient vector space.
    (n=4)
    Dimension: 12
    Rank of K0(X) (max. collection length): 32
    Fano index (max. orbit length): 5

    cM1 is exceptional.

    Can candidate be embedded in the Lefschetz collection LC1?
       Not after row 0.
       Not after row 1.
       Not after row 2.
       Not after row 3.
       Yes after row 4 to the columns [0, 1, 2, 3, 4].
       Yes after row 5 to the columns [0, 1, 2, 3, 4].
       Yes after row 6 to the columns [0, 1, 2, 3, 4].

    Can candidate be embedded in the Lefschetz collection LC2?
       Not after row 0.
       Not after row 1.
       Not after row 2.
       Not after row 3.
       Yes after row 4 to the columns [0, 1, 2, 3, 4].
       Yes after row 5 to the columns [0, 1, 2, 3, 4].
       Yes after row 6 to the columns [0, 1, 2, 3, 4].

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

    cM1 is exceptional.

    Can candidate be embedded in the Lefschetz collection LC1?
       Not after row 0.
       Not after row 1.
       Not after row 2.
       Not after row 3.
       Not after row 4.
       Not after row 5.
       Not after row 6.
       Not after row 7.
       Yes after row 8 to the columns [0, 1, 2, 3, 4, 5, 6].
       Yes after row 9 to the columns [0, 1, 2, 3, 4, 5, 6].
       Yes after row 10 to the columns [0, 1, 2, 3, 4, 5, 6].
       Yes after row 11 to the columns [0, 1, 2, 3, 4, 5, 6].

    Can candidate be embedded in the Lefschetz collection LC2?
       Not after row 0.
       Not after row 1.
       Not after row 2.
       Not after row 3.
       Not after row 4.
       Not after row 5.
       Not after row 6.
       Not after row 7.
       Not after row 8.
       Not after row 9.
       Yes after row 10 to the columns [0, 1, 2, 3, 4, 5, 6].
       Yes after row 11 to the columns [0, 1, 2, 3, 4, 5, 6].

X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 13-dimensional ambient vector space.
    (n=6)
    Dimension: 24
    Rank of K0(X) (max. collection length): 160
    Fano index (max. orbit length): 9

    cM1 is exceptional.

    Can candidate be embedded in the Lefschetz collection LC1?
       Not after row 0.
       Not after row 1.
       Not after row 2.
       Not after row 3.
       Not after row 4.
       Not after row 5.
       Not after row 6.
       Not after row 7.
       Not after row 8.
       Not after row 9.

k = 3

for n in [ 4 .. 10 ] :
    N = 2*n+1
    X = Orthogonal_Grassmannian(k,N)
    G = X.Parent_Group()
    fw = X.Basis('fw')

    print( 'X: '+str(X) )
    print( 3*' ' , '(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()

    if n == 4 : cR = X.calU( 2*fw[4] )(-1)
    else      : cR = X.calU(   fw[4] )(-1)

    cM2 = X.calU().Dual()*cR.Wedge(n-3) + cR.Wedge(n-3) + cR.Wedge(n-4)
    cM1 = cM2*-1
    for Smd in ( X.calS()*G.rmV( fw[n] ) ).Irreducible_Components() :
        cM1 += Smd
        
    print(cM1)

X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 9-dimensional ambient vector space.
    (n=4)
    Dimension: 12
    Rank of K0(X) (max. collection length): 32
    Fano index (max. orbit length): 5

VB(Lambda[2] + 2*Lambda[4])(-1) + VB(Lambda[2]) + VB(Lambda[1]) + VB(2*Lambda[4]) + VB(0)(1)
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

VB(Lambda[2] + 2*Lambda[5])(-1) + VB(Lambda[2] + Lambda[4])(-1) + VB(Lambda[2]) + VB(Lambda[1] + Lambda[4])(-1) + VB(Lambda[1]) + VB(2*Lambda[5]) + VB(Lambda[4]) + VB(0)(1) + VB(0)
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 13-dimensional ambient vector space.
    (n=6)
    Dimension: 24
    Rank of K0(X) (max. collection length): 160
    Fano index (max. orbit length): 9

VB(Lambda[2] + 2*Lambda[6])(-1) + VB(Lambda[2] + Lambda[5])(-1) + VB(Lambda[2] + Lambda[4])(-1) + VB(Lambda[2]) + VB(Lambda[1] + Lambda[5])(-1) + VB(Lambda[1] + Lambda[4])(-1) + VB(Lambda[1]) + VB(2*Lambda[6]) + VB(Lambda[5]) + VB(Lambda[4]) + VB(0)(1) + VB(Lambda[4])(-1) + VB(0)
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 15-dimensional ambient vector space.
    (n=7)
    Dimension: 30
    Rank of K0(X) (max. collection length): 280
    Fano index (max. orbit length): 11

VB(Lambda[2] + 2*Lambda[7])(-1) + VB(Lambda[2] + Lambda[6])(-1) + VB(Lambda[2] + Lambda[5])(-1) + VB(Lambda[2] + Lambda[4])(-1) + VB(Lambda[2]) + VB(Lambda[1] + Lambda[6])(-1) + VB(Lambda[1] + Lambda[5])(-1) + VB(Lambda[1] + Lambda[4])(-1) + VB(Lambda[1]) + VB(2*Lambda[7]) + VB(Lambda[6]) + VB(Lambda[5]) + VB(Lambda[4]) + VB(0)(1) + VB(Lambda[5])(-1) + VB(Lambda[4])(-1) + VB(0)
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 17-dimensional ambient vector space.
    (n=8)
    Dimension: 36
    Rank of K0(X) (max. collection length): 448
    Fano index (max. orbit length): 13

VB(Lambda[2] + 2*Lambda[8])(-1) + VB(Lambda[2] + Lambda[7])(-1) + VB(Lambda[2] + Lambda[6])(-1) + VB(Lambda[2] + Lambda[5])(-1) + VB(Lambda[2] + Lambda[4])(-1) + VB(Lambda[2]) + VB(Lambda[1] + Lambda[7])(-1) + VB(Lambda[1] + Lambda[6])(-1) + VB(Lambda[1] + Lambda[5])(-1) + VB(Lambda[1] + Lambda[4])(-1) + VB(Lambda[1]) + VB(2*Lambda[8]) + VB(Lambda[7]) + VB(Lambda[6]) + VB(Lambda[5]) + VB(Lambda[4]) + VB(0)(1) + VB(Lambda[6])(-1) + VB(Lambda[5])(-1) + VB(Lambda[4])(-1) + VB(0)
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 19-dimensional ambient vector space.
    (n=9)
    Dimension: 42
    Rank of K0(X) (max. collection length): 672
    Fano index (max. orbit length): 15

VB(Lambda[2] + 2*Lambda[9])(-1) + VB(Lambda[2] + Lambda[8])(-1) + VB(Lambda[2] + Lambda[7])(-1) + VB(Lambda[2] + Lambda[6])(-1) + VB(Lambda[2] + Lambda[5])(-1) + VB(Lambda[2] + Lambda[4])(-1) + VB(Lambda[2]) + VB(Lambda[1] + Lambda[8])(-1) + VB(Lambda[1] + Lambda[7])(-1) + VB(Lambda[1] + Lambda[6])(-1) + VB(Lambda[1] + Lambda[5])(-1) + VB(Lambda[1] + Lambda[4])(-1) + VB(Lambda[1]) + VB(2*Lambda[9]) + VB(Lambda[8]) + VB(Lambda[7]) + VB(Lambda[6]) + VB(Lambda[5]) + VB(Lambda[4]) + VB(0)(1) + VB(Lambda[7])(-1) + VB(Lambda[6])(-1) + VB(Lambda[5])(-1) + VB(Lambda[4])(-1) + VB(0)
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 21-dimensional ambient vector space.
    (n=10)
    Dimension: 48
    Rank of K0(X) (max. collection length): 960
    Fano index (max. orbit length): 17

VB(Lambda[2] + 2*Lambda[10])(-1) + VB(Lambda[2] + Lambda[9])(-1) + VB(Lambda[2] + Lambda[8])(-1) + VB(Lambda[2] + Lambda[7])(-1) + VB(Lambda[2] + Lambda[6])(-1) + VB(Lambda[2] + Lambda[5])(-1) + VB(Lambda[2] + Lambda[4])(-1) + VB(Lambda[2]) + VB(Lambda[1] + Lambda[9])(-1) + VB(Lambda[1] + Lambda[8])(-1) + VB(Lambda[1] + Lambda[7])(-1) + VB(Lambda[1] + Lambda[6])(-1) + VB(Lambda[1] + Lambda[5])(-1) + VB(Lambda[1] + Lambda[4])(-1) + VB(Lambda[1]) + VB(2*Lambda[10]) + VB(Lambda[9]) + VB(Lambda[8]) + VB(Lambda[7]) + VB(Lambda[6]) + VB(Lambda[5]) + VB(Lambda[4]) + VB(0)(1) + VB(Lambda[8])(-1) + VB(Lambda[7])(-1) + VB(Lambda[6])(-1) + VB(Lambda[5])(-1) + VB(Lambda[4])(-1) + VB(0)