Object_cM.ipynb

Kernel: SageMath 9.8

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

k = 3
n = 4

N = 2*n+1
X = Orthogonal_Grassmannian(k,N)
G = X.Parent_Group()
fw = X.Basis('fw')

Output = dict({})
Output.update({ 1 : '' })
Output[1] += 'X: '+str(X)+'\n'
Output[1] += '(n='+str(n)+')'+'\n'
d = X.Dimension()
Output[1] += 3*' '+'Dimension: '+str(d)+'\n'
lMax = X.K0().rank()
Output[1] += 3*' '+'Rank of K0(X) (max. collection length): '+str(lMax)+'\n'
wMax = X.Fano_Index()
Output[1] += 3*' '+'Fano index (max. orbit length): '+str(wMax)+'\n'
Output[1] += '\n'

Output.update({ 2 : '' })
Output[2] += 3*' '+'Consecutive Lefschetz collection.'+'\n'
LC1 = X.My_Collection( Modus='Con' )
l1 = len(LC1)
Output[2] += 3*' '+'Starting block:'+'\n'
w1_0 = len( LC1.Starting_Block() )
s = ceil( math.log10( w1_0 ) )
for i , cE in enumerate ( LC1.Starting_Block() , start=1 ) :
    Output[2] += 6*' '+(s-floor(math.log10(i))-1)*' '+str(i)+' '+str(cE)+'\n'
Output[2] += '\n'

Output.update({ 3 : '' })
Output[3] += 3*' '+'Alternating Lefschetz collection.'+'\n'
LC2 = X.My_Collection( Modus='Alt' )
l2 = len(LC2)
Output[3] += 3*' '+'Starting block:'+'\n'
w2_0 = len( LC2.Starting_Block() )
s = ceil( math.log10( w2_0 ) )
for i , cE in enumerate ( LC2.Starting_Block() , start=1 ) :
    Output[3] += 6*' '+(s-floor(math.log10(i))-1)*' '+str(i)+' '+str(cE)+'\n'
Output[3] += '\n'

for Key , Line in Output.items() :
    print( Line )

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


   Consecutive Lefschetz collection.
   Starting block:
      1 VB(0)
      2 VB(Lambda[1])
      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])


   Alternating Lefschetz collection.
   Starting block:
      1 VB(0)
      2 VB(Lambda[4])
      3 VB(Lambda[1])
      4 VB(Lambda[2])
      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])

cR = X.calU( 2*fw[4] )(-1)

A = X.calU().Dual().Wedge(2)
B = X.calU() * cR
print( A.EXT(B) )

{}

cM1 = X.calU( fw[1]+2*fw[4] )(-1) + X.calU( 2*fw[4] )(-1) + X.calU( fw[n-1] )(-1)
#cM2 = X.Complex({ -2 : cM1 , -1 : X.calU( fw[n] ) * G.rmV( fw[n] ) , 0 : 'Cokernel' }).SemiSimplification(0)
cM2 = cM1*-1
for Smd in ( X.calU( fw[n] ) * G.rmV( fw[n] ) ).Irreducible_Components() :
    cM2 += Smd

print( 'cM2 has semi-simplification '+bcolors.OKBLUE+str(cM2)+bcolors.ENDC+'.' )
print()

for i , LC in enumerate([ LC1 , LC2 ] , start=1 ) :
        print( 3*' ' , 'Can candidate be embedded in the Lefschetz collection LC'+str(i)+'?' )
        print()

        for xPos , Admissible_Columns in LC.Test_For_Extension ( New_Object=cM2 , 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()

cM2 has semi-simplification VB(Lambda[2] - Lambda[3] + 2*Lambda[4]) + VB(Lambda[2]) + VB(Lambda[1]) + VB(2*Lambda[4]) + VB(Lambda[3]).

    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].

rV = G.rmV( fw[1] )
Cplx = X.Complex({ 0 : 'Kernel' ,
                   1 : X.calU().Dual().Symmetric_Power(3)(1) * rV.exterior_power(2) ,
                   2 : X.calU().Dual().Symmetric_Power(4)(1) * rV.exterior_power(1) ,
                   3 : X.calU().Dual().Symmetric_Power(5)(1) * rV.exterior_power(0)
                  })
for E in Cplx.SemiSimplification(0) :
    print(E)

VB(3*Lambda[1] + Lambda[3])
VB(2*Lambda[1] + Lambda[2] + 2*Lambda[4])
VB(3*Lambda[1] + Lambda[2] - Lambda[3] + 2*Lambda[4])
VB(2*Lambda[1] + 2*Lambda[4])
VB(2*Lambda[1] + 2*Lambda[2])
VB(Lambda[1] + Lambda[2] + Lambda[3])
VB(2*Lambda[1] + 2*Lambda[3])
VB(3*Lambda[1] + 2*Lambda[4])
VB(4*Lambda[1])
VB(2*Lambda[1] + Lambda[2])

cM2 = X.calU( fw[1]+2*fw[4] )(-1) + X.calU( 2*fw[4] )(-1) + X.calU( fw[n-1] )(-1)
cM3 = cM2*-1
for Smd in ( X.calU( fw[n] ) * G.rmV( fw[n] ) ).Irreducible_Components() :
    cM3 += Smd
cM = cM3