Object_cM2.ipynb

Kernel: SageMath 10.3

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

k = 3
n = 6

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 13-dimensional ambient vector space.
(n=6)
   Dimension: 24
   Rank of K0(X) (max. collection length): 160
   Fano index (max. orbit length): 9


   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(3*Lambda[1])
       8 VB(2*Lambda[1] + Lambda[2])
       9 VB(Lambda[1] + 2*Lambda[2])
      10 VB(3*Lambda[2])
      11 VB(3*Lambda[1] + Lambda[2])
      12 VB(2*Lambda[1] + 2*Lambda[2])
      13 VB(Lambda[6])
      14 Equivariant extension of VB(Lambda[1] + Lambda[6]) by VB(Lambda[6])
      15 Equivariant extension of VB(2*Lambda[1] + Lambda[6]) by VB(Lambda[1] + Lambda[6])
      16 Equivariant extension of VB(3*Lambda[1] + Lambda[6]) by VB(2*Lambda[1] + Lambda[6])
      17 Equivariant extension of VB(4*Lambda[1] + Lambda[6]) by VB(3*Lambda[1] + Lambda[6])


   Alternating Lefschetz collection.
   Starting block:
       1 VB(0)
       2 VB(Lambda[6])
       3 VB(Lambda[1])
       4 VB(Lambda[2])
       5 Equivariant extension of VB(Lambda[1] + Lambda[6]) by VB(Lambda[6])
       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[6]) by VB(Lambda[1] + Lambda[6])
      10 VB(3*Lambda[1])
      11 VB(2*Lambda[1] + Lambda[2])
      12 VB(Lambda[1] + 2*Lambda[2])
      13 VB(3*Lambda[2])
      14 Equivariant extension of VB(3*Lambda[1] + Lambda[6]) by VB(2*Lambda[1] + Lambda[6])
      15 VB(3*Lambda[1] + Lambda[2])
      16 VB(2*Lambda[1] + 2*Lambda[2])
      17 Equivariant extension of VB(4*Lambda[1] + Lambda[6]) by VB(3*Lambda[1] + Lambda[6])

A1 = X.calU( 3*fw[1]+fw[n-1] )(-1)
A2 = X.calU( 2*fw[1]+fw[n-2] )(-1)
A3 = X.calU(         fw[n-1] )(-1)

#print( A1.EXT(A2) )
#print( A1.EXT(A3) )
#print( A2.EXT(A1) )
#print( A2.EXT(A3) )
#print( A3.EXT(A1) )
#print( A3.EXT(A2) )

E = X.calU( 0*fw[n] ) * G.rmV( fw[4] )
print( E )
for Smd in ( E ).Irreducible_Components() :
    print( 3*' ' , Smd)


#cM1 = X.calU( fw[1]+2*fw[n] )(-1) + X.calU( 2*fw[n] )(-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
#
#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()

B6(0,0,0,1,0,0) * VB(0)
    VB(Lambda[1] + Lambda[2] - 2*Lambda[3] + Lambda[5])
    VB(Lambda[1] - Lambda[3] + 2*Lambda[6])
    VB(Lambda[2] - 2*Lambda[3] + 2*Lambda[6])
    VB(Lambda[2] - Lambda[3] + Lambda[5])
    VB(Lambda[1] - 2*Lambda[3] + Lambda[5])
    VB(2*Lambda[2] - 2*Lambda[3] + Lambda[4])
    VB(2*Lambda[1] - 2*Lambda[3] + Lambda[4])
    VB(-Lambda[3] + 2*Lambda[6])
    VB(Lambda[1] - Lambda[3] + Lambda[4])
    VB(-Lambda[3] + Lambda[5])
    VB(Lambda[2] - 2*Lambda[3] + Lambda[4])
    VB(Lambda[1] + Lambda[2] - Lambda[3])
    VB(Lambda[4])
    VB(-2*Lambda[3] + Lambda[4])
    VB(Lambda[2])
    VB(Lambda[1] - Lambda[3])
    VB(0)

print( G.rmV( fw[6] ).exterior_power(4) )

2*B6(0,0,0,0,0,0) + B6(1,0,0,0,0,0) + B6(0,0,1,0,0,0) + 2*B6(0,0,0,1,0,0) + 2*B6(0,0,0,0,1,0) + B6(0,0,0,0,0,2) + B6(2,0,0,0,0,0) + B6(1,0,0,1,0,0) + B6(1,0,0,0,1,0) + B6(0,1,0,0,0,2) + B6(0,0,2,0,0,0) + B6(0,0,1,1,0,0) + B6(0,0,1,0,0,2) + B6(0,0,0,2,0,0)

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