Playground.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 = 4

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

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

E = X.calU().Dual() * cR

print( 'cU.Dual * cR = '+str( E ) )
print( '-> rank = '+str(E.Rank()) )
print()

print( 'Wedge^9 E = '+str(E.Wedge(9)) )
print( 'Wedge^3 cU.Dual = '+str(X.calU().Dual().Wedge(3)) )
print( 'Wedge^3 cR = '+str(cR.Wedge(3)) )

cU.Dual * cR = VB(Lambda[1] + 2*Lambda[4])(-1)
-> rank = 9

Wedge^9 E = VB(0)(3)
Wedge^3 cU.Dual = VB(0)(1)
Wedge^3 cR = VB(0)

print( 'Irreducible components of trivial vector bundle rV*cO:' )
for Smd in X.Trivial_Vector_Bundle().Irreducible_Components() :
    print( 3*' ' , Smd )
print()

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

cQ = X.calU().Dual().Extend_Equivariantly_By( cR )
print( 'cQ:' , cQ )
print()

cT_ss = X.calU().Dual()*cQ.SemiSimplification() - X.calU().Dual().Symmetric_Power(2)
print( 'Semi-simplification of cT:' , cT_ss )
print()

A = X.calU().Dual().Wedge(2)
print( 'A='+str(A) )
B = X.calU().Dual() * cR
print( 'B='+str(B) )
print( 'EXT(A,B) = '+str(A.EXT(B)) )

Irreducible components of trivial vector bundle rV*cO:
    VB(Lambda[1])
    VB(2*Lambda[4])(-1)
    VB(Lambda[2])(-1)

cQ: Equivariant extension of VB(Lambda[1]) by VB(2*Lambda[4])(-1)

Semi-simplification of cT: VB(Lambda[2]) + VB(Lambda[1] + 2*Lambda[4])(-1)

A=VB(Lambda[2])
B=VB(Lambda[1] + 2*Lambda[4])(-1)
EXT(A,B) = {1: B4(0,0,0,0)}