Filtration.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 = 5

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 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])

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

print( G.rmV( fw[n] ).exterior_power(2) )

VB(Lambda[2] - Lambda[3] + Lambda[5])
VB(Lambda[1] - Lambda[3] + Lambda[5])
VB(Lambda[5])
VB(-Lambda[3] + Lambda[5])

B5(0,0,0,0,0) + B5(0,0,1,0,0) + B5(0,0,0,1,0)

print( X.calS().Wedge(2) )

VB(Lambda[4]) + VB(Lambda[3])

cUˇ

print( '' )

cUˇ

cR = X.calU( fw[4] )(-1)
for i in [ 0 .. cR.Rank() ] :
    print( i , cR.Wedge(i) )

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