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Environment to perform calculations of equivariant vector bundles on homogeneous varieties

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License: GPL3
ubuntu2204
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Kernel: SageMath 10.3
%run '/home/user/Equivariant_Vector_Bundles_On_Homogeneous_Varieties__0-2/tests/Initialize.ipynb'

from IPython.display import clear_output , display , HTML

from Equivariant_Vector_Bundles_On_Homogeneous_Varieties.Base_Space.Orthogonal_Grassmannian import Orthogonal_Grassmannian

k = 2
for n in [ 3 .. 10 ] :
    N = 2*n+1
    X = Orthogonal_Grassmannian(k,N)
    print( 'X:' , X )
    n = floor(N/2)
    print( 3*' ' , 'n='+str(n) )
    d = X.Dimension()
    print( 3*' ' , 'd='+str(d) )
    FanoIndex = X.Fano_Index()
    print( 3*' ' , 'FanoIndex='+str(FanoIndex) )
    r = floor(d/2)
    print( 3*' ' , 'r='+str(r) )
    print()

    sr = X.Basis('sr')
    fw = X.Basis('fw')

    if n == k+1 : cR = X.calU( 2*fw[k+1] )(-1)
    else        : cR = X.calU(   fw[k+1] )(-1)
    print( 3*' ' , 'cR:' , cR )
    print()
    
    #cQ = X.calU().Dual().Extend_Equivariantly_By( cR )
    #print( 3*' ' , 'cQ:' , cQ )
    #print()

    Body = []
    for Smd in ( X.calU().Dual() * cR ).Wedge(3)(-1) :
        Body += [ [ Smd , Smd.Highest_Weight().to_ambient() , Smd.Cohomology() ] ]
    for Row in str(table( Body , frame=True )).split('\n') :
        print( 3*' ' , Row )
    print()
X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 7-dimensional ambient vector space. n=3 d=7 FanoIndex=4 r=3 cR: VB(2*Lambda[3])(-1) ┌─────────────────────────────────┬────────────┬────────────────┐ │ VB(Lambda[1] + 4*Lambda[3])(-2) | (1, 0, 2) | {1: B3(0,0,2)} | ├─────────────────────────────────┼────────────┼────────────────┤ │ VB(3*Lambda[1])(-1) | (2, -1, 0) | {} | ├─────────────────────────────────┼────────────┼────────────────┤ │ VB(Lambda[1] + 2*Lambda[3])(-1) | (1, 0, 1) | {} | └─────────────────────────────────┴────────────┴────────────────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 9-dimensional ambient vector space. n=4 d=11 FanoIndex=6 r=5 cR: VB(Lambda[3])(-1) ┌─────────────────────────────────────────────┬───────────────┬──────────────────┐ │ VB(Lambda[1] + Lambda[3] + 2*Lambda[4])(-2) | (1, 0, 2, 1) | {1: B4(0,0,0,2)} | ├─────────────────────────────────────────────┼───────────────┼──────────────────┤ │ VB(3*Lambda[1] + 2*Lambda[4])(-2) | (2, -1, 1, 1) | {} | ├─────────────────────────────────────────────┼───────────────┼──────────────────┤ │ VB(Lambda[1] + Lambda[3])(-1) | (1, 0, 1, 0) | {} | └─────────────────────────────────────────────┴───────────────┴──────────────────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 11-dimensional ambient vector space. n=5 d=15 FanoIndex=8 r=7 cR: VB(Lambda[3])(-1) ┌───────────────────────────────────────────┬──────────────────┬────────────────────┐ │ VB(Lambda[1] + Lambda[3] + Lambda[4])(-2) | (1, 0, 2, 1, 0) | {1: B5(0,0,0,1,0)} | ├───────────────────────────────────────────┼──────────────────┼────────────────────┤ │ VB(3*Lambda[1] + 2*Lambda[5])(-2) | (2, -1, 1, 1, 1) | {} | ├───────────────────────────────────────────┼──────────────────┼────────────────────┤ │ VB(Lambda[1] + Lambda[3])(-1) | (1, 0, 1, 0, 0) | {} | └───────────────────────────────────────────┴──────────────────┴────────────────────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 13-dimensional ambient vector space. n=6 d=19 FanoIndex=10 r=9 cR: VB(Lambda[3])(-1) ┌───────────────────────────────────────────┬─────────────────────┬──────────────────────┐ │ VB(Lambda[1] + Lambda[3] + Lambda[4])(-2) | (1, 0, 2, 1, 0, 0) | {1: B6(0,0,0,1,0,0)} | ├───────────────────────────────────────────┼─────────────────────┼──────────────────────┤ │ VB(3*Lambda[1] + Lambda[5])(-2) | (2, -1, 1, 1, 1, 0) | {} | ├───────────────────────────────────────────┼─────────────────────┼──────────────────────┤ │ VB(Lambda[1] + Lambda[3])(-1) | (1, 0, 1, 0, 0, 0) | {} | └───────────────────────────────────────────┴─────────────────────┴──────────────────────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 15-dimensional ambient vector space. n=7 d=23 FanoIndex=12 r=11 cR: VB(Lambda[3])(-1) ┌───────────────────────────────────────────┬────────────────────────┬────────────────────────┐ │ VB(Lambda[1] + Lambda[3] + Lambda[4])(-2) | (1, 0, 2, 1, 0, 0, 0) | {1: B7(0,0,0,1,0,0,0)} | ├───────────────────────────────────────────┼────────────────────────┼────────────────────────┤ │ VB(3*Lambda[1] + Lambda[5])(-2) | (2, -1, 1, 1, 1, 0, 0) | {} | ├───────────────────────────────────────────┼────────────────────────┼────────────────────────┤ │ VB(Lambda[1] + Lambda[3])(-1) | (1, 0, 1, 0, 0, 0, 0) | {} | └───────────────────────────────────────────┴────────────────────────┴────────────────────────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 17-dimensional ambient vector space. n=8 d=27 FanoIndex=14 r=13 cR: VB(Lambda[3])(-1) ┌───────────────────────────────────────────┬───────────────────────────┬──────────────────────────┐ │ VB(Lambda[1] + Lambda[3] + Lambda[4])(-2) | (1, 0, 2, 1, 0, 0, 0, 0) | {1: B8(0,0,0,1,0,0,0,0)} | ├───────────────────────────────────────────┼───────────────────────────┼──────────────────────────┤ │ VB(3*Lambda[1] + Lambda[5])(-2) | (2, -1, 1, 1, 1, 0, 0, 0) | {} | ├───────────────────────────────────────────┼───────────────────────────┼──────────────────────────┤ │ VB(Lambda[1] + Lambda[3])(-1) | (1, 0, 1, 0, 0, 0, 0, 0) | {} | └───────────────────────────────────────────┴───────────────────────────┴──────────────────────────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 19-dimensional ambient vector space. n=9 d=31 FanoIndex=16 r=15 cR: VB(Lambda[3])(-1) ┌───────────────────────────────────────────┬──────────────────────────────┬────────────────────────────┐ │ VB(Lambda[1] + Lambda[3] + Lambda[4])(-2) | (1, 0, 2, 1, 0, 0, 0, 0, 0) | {1: B9(0,0,0,1,0,0,0,0,0)} | ├───────────────────────────────────────────┼──────────────────────────────┼────────────────────────────┤ │ VB(3*Lambda[1] + Lambda[5])(-2) | (2, -1, 1, 1, 1, 0, 0, 0, 0) | {} | ├───────────────────────────────────────────┼──────────────────────────────┼────────────────────────────┤ │ VB(Lambda[1] + Lambda[3])(-1) | (1, 0, 1, 0, 0, 0, 0, 0, 0) | {} | └───────────────────────────────────────────┴──────────────────────────────┴────────────────────────────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 21-dimensional ambient vector space. n=10 d=35 FanoIndex=18 r=17 cR: VB(Lambda[3])(-1) ┌───────────────────────────────────────────┬─────────────────────────────────┬───────────────────────────────┐ │ VB(Lambda[1] + Lambda[3] + Lambda[4])(-2) | (1, 0, 2, 1, 0, 0, 0, 0, 0, 0) | {1: B10(0,0,0,1,0,0,0,0,0,0)} | ├───────────────────────────────────────────┼─────────────────────────────────┼───────────────────────────────┤ │ VB(3*Lambda[1] + Lambda[5])(-2) | (2, -1, 1, 1, 1, 0, 0, 0, 0, 0) | {} | ├───────────────────────────────────────────┼─────────────────────────────────┼───────────────────────────────┤ │ VB(Lambda[1] + Lambda[3])(-1) | (1, 0, 1, 0, 0, 0, 0, 0, 0, 0) | {} | └───────────────────────────────────────────┴─────────────────────────────────┴───────────────────────────────┘ 


k = 2
for n in [ 4 .. 10 ] :
    N = 2*n
    X = Orthogonal_Grassmannian(k,N)
    print( 'X:' , X )
    n = floor(N/2)
    print( 3*' ' , 'n='+str(n) )
    d = X.Dimension()
    print( 3*' ' , 'd='+str(d) )
    FanoIndex = X.Fano_Index()
    print( 3*' ' , 'FanoIndex='+str(FanoIndex) )
    r = floor(d/2)
    print( 3*' ' , 'r='+str(r) )
    print()

    sr = X.Basis('sr')
    fw = X.Basis('fw')

    if n == k+2 : cR = X.calU( fw[k+1] + fw[k+2] )(-1)
    else        : cR = X.calU( fw[k+1] )(-1)
    print( 3*' ' , 'cR:' , cR )
    print()
    
    #cQ = X.calU().Dual().Extend_Equivariantly_By( cR )
    #print( 3*' ' , 'cQ:' , cQ )
    #print()

    Body = []
    for Smd in ( X.calU().Dual() * cR ).Wedge(2) :
        #if X.Is_Levi_Dominant( Smd.Highest_Weight() ) : Body += [ [ Smd , Smd.Highest_Weight().to_ambient() , Smd.Cohomology() ] ]
        #else                                          :
        if n in [ 5 ] : Body += [ [ Smd , Smd.Highest_Weight().to_ambient() , '?' ] ]
        else          : Body += [ [ Smd , Smd.Highest_Weight().to_ambient() , Smd.Cohomology() ] ]
    for Row in str(table( Body , frame=True )).split('\n') :
        print( 3*' ' , Row )
    print()
X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 8-dimensional ambient vector space. n=4 d=9 FanoIndex=5 r=4 cR: VB(Lambda[3] + Lambda[4])(-1) ┌───────────────────────────────────┬───────────────┬──────────────────┐ │ VB(2*Lambda[3] + 2*Lambda[4])(-1) | (1, 1, 2, 0) | {} | ├───────────────────────────────────┼───────────────┼──────────────────┤ │ VB(2*Lambda[1] + 2*Lambda[4])(-1) | (2, 0, 1, 1) | {} | ├───────────────────────────────────┼───────────────┼──────────────────┤ │ VB(2*Lambda[1] + 2*Lambda[3])(-1) | (2, 0, 1, -1) | {} | ├───────────────────────────────────┼───────────────┼──────────────────┤ │ VB(0)(1) | (1, 1, 0, 0) | {0: D4(0,1,0,0)} | └───────────────────────────────────┴───────────────┴──────────────────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 10-dimensional ambient vector space. n=5 d=13 FanoIndex=7 r=6 cR: VB(Lambda[3])(-1) ┌───────────────────────────────────┬────────────────────────┬───┐ │ VB(2*Lambda[3])(-1) | (1, 1, 2, 0, 0) | ? | ├───────────────────────────────────┼────────────────────────┼───┤ │ VB(2*Lambda[1] + Lambda[4])(-1/2) | (2, 0, 1/2, 1/2, -1/2) | ? | └───────────────────────────────────┴────────────────────────┴───┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 12-dimensional ambient vector space. n=6 d=17 FanoIndex=9 r=8 cR: VB(Lambda[3])(-1) ┌─────────────────────────────────┬────────────────────┬────┐ │ VB(2*Lambda[3])(-1) | (1, 1, 2, 0, 0, 0) | {} | ├─────────────────────────────────┼────────────────────┼────┤ │ VB(2*Lambda[1] + Lambda[4])(-1) | (2, 0, 1, 1, 0, 0) | {} | └─────────────────────────────────┴────────────────────┴────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 14-dimensional ambient vector space. n=7 d=21 FanoIndex=11 r=10 cR: VB(Lambda[3])(-1) ┌─────────────────────────────────┬───────────────────────┬────┐ │ VB(2*Lambda[3])(-1) | (1, 1, 2, 0, 0, 0, 0) | {} | ├─────────────────────────────────┼───────────────────────┼────┤ │ VB(2*Lambda[1] + Lambda[4])(-1) | (2, 0, 1, 1, 0, 0, 0) | {} | └─────────────────────────────────┴───────────────────────┴────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 16-dimensional ambient vector space. n=8 d=25 FanoIndex=13 r=12 cR: VB(Lambda[3])(-1) ┌─────────────────────────────────┬──────────────────────────┬────┐ │ VB(2*Lambda[3])(-1) | (1, 1, 2, 0, 0, 0, 0, 0) | {} | ├─────────────────────────────────┼──────────────────────────┼────┤ │ VB(2*Lambda[1] + Lambda[4])(-1) | (2, 0, 1, 1, 0, 0, 0, 0) | {} | └─────────────────────────────────┴──────────────────────────┴────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 18-dimensional ambient vector space. n=9 d=29 FanoIndex=15 r=14 cR: VB(Lambda[3])(-1) ┌─────────────────────────────────┬─────────────────────────────┬────┐ │ VB(2*Lambda[3])(-1) | (1, 1, 2, 0, 0, 0, 0, 0, 0) | {} | ├─────────────────────────────────┼─────────────────────────────┼────┤ │ VB(2*Lambda[1] + Lambda[4])(-1) | (2, 0, 1, 1, 0, 0, 0, 0, 0) | {} | └─────────────────────────────────┴─────────────────────────────┴────┘ X: Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 20-dimensional ambient vector space. n=10 d=33 FanoIndex=17 r=16 cR: VB(Lambda[3])(-1) ┌─────────────────────────────────┬────────────────────────────────┬────┐ │ VB(2*Lambda[3])(-1) | (1, 1, 2, 0, 0, 0, 0, 0, 0, 0) | {} | ├─────────────────────────────────┼────────────────────────────────┼────┤ │ VB(2*Lambda[1] + Lambda[4])(-1) | (2, 0, 1, 1, 0, 0, 0, 0, 0, 0) | {} | └─────────────────────────────────┴────────────────────────────────┴────┘ 


k = 2
for n in [ 4 .. 10 ] :
    G = Irreducible_Cartan_Group( 'D' , n )
    P = G.Maximal_Parabolic_Subgroup(k)
    X = G/P
    print( 'X:' , X )
    n = floor(N/2)
    print( 3*' ' , 'n='+str(n) )
    d = X.Dimension()
    print( 3*' ' , 'd='+str(d) )
    #FanoIndex = X.Fano_Index()
    #print( 3*' ' , 'FanoIndex='+str(FanoIndex) )
    r = floor(d/2)
    print( 3*' ' , 'r='+str(r) )
    print()

    sr = X.Basis('sr')
    fw = X.Basis('fw')

    #if n == k+1 : cR = X.calU( 2*fw[k+1] )(-1)
    #else        : cR = X.calU(   fw[k+1] )(-1)
    #print( 3*' ' , 'cR:' , cR )
    #print()
    
    #cQ = X.calU().Dual().Extend_Equivariantly_By( cR )
    #print( 3*' ' , 'cQ:' , cQ )
    #print()

    #Body = []
    #for Smd in ( X.calU().Dual() * cR ).Wedge(2) :
    #    Body += [ [ Smd , Smd.Highest_Weight().to_ambient() , Smd.Cohomology() ] ]
    #for Row in str(table( Body , frame=True )).split('\n') :
    #    print( 3*' ' , Row )
    #print()

    E = X.Trivial_Vector_Bundle()
    print( 3*' ' , 'E = '+str(E) )
    print( 3*' ' , 'E.Rank() = '+str(E.Rank()) )
    print()

    Stock = X.Zero_Vector_Bundle()
    for Factor , Weyl_Character in E.Summands() :
        for Weight , Multiplicity in Weyl_Character.weight_multiplicities().items() :
            Weight = Weight.to_weight_space()
            if X.Is_Levi_Dominant( Weight ) :
                Stock += X.calU( Weight ) * Multiplicity * Factor
    print( 3*' ' , 'Stock = '+str(Stock) )
    print( 3*' ' , 'Stock.Rank() = '+str(Stock.Rank()) )
    print()
    
    for Irreducible_Summand in Stock :
        print( 3*' ' , Irreducible_Summand )
        for Weight , Multiplicity in Irreducible_Summand.Weights() :
            if X.Is_Levi_Dominant( Weight ) and X.Compare_Weights( Irreducible_Summand.Highest_Weight() , Weight ) == '>' :
                print( 6*' ' , str(Multiplicity)+'x'+str(Weight) )
    print()
X: Smooth projective variety D4/P({1, 3, 4}). n=10 d=9 r=4 E = D4(1,0,0,0) * VB(0) E.Rank() = 8 Stock = VB(Lambda[1]) + VB(Lambda[3] + Lambda[4])(-1) + VB(Lambda[1])(-1) Stock.Rank() = 8 VB(Lambda[1]) VB(Lambda[3] + Lambda[4])(-1) VB(Lambda[1])(-1) X: Smooth projective variety D5/P({1, 3, 4, 5}). n=10 d=13 r=6 E = D5(1,0,0,0,0) * VB(0) E.Rank() = 10 Stock = VB(Lambda[1]) + VB(Lambda[3])(-1) + VB(Lambda[1])(-1) Stock.Rank() = 7 VB(Lambda[1]) VB(Lambda[3])(-1) VB(Lambda[1])(-1) X: Smooth projective variety D6/P({1, 3, 4, 5, 6}). n=10 d=17 r=8 E = D6(1,0,0,0,0,0) * VB(0) E.Rank() = 12 Stock = VB(Lambda[1]) + VB(Lambda[3])(-1) + VB(Lambda[1])(-1) Stock.Rank() = 8 VB(Lambda[1]) VB(Lambda[3])(-1) VB(Lambda[1])(-1) X: Smooth projective variety D7/P({1, 3, 4, 5, 6, 7}). n=10 d=21 r=10 E = D7(1,0,0,0,0,0,0) * VB(0) E.Rank() = 14 Stock = VB(Lambda[1]) + VB(Lambda[3])(-1) + VB(Lambda[1])(-1) Stock.Rank() = 9 VB(Lambda[1]) VB(Lambda[3])(-1) VB(Lambda[1])(-1) X: Smooth projective variety D8/P({1, 3, 4, 5, 6, 7, 8}). n=10 d=25 r=12 E = D8(1,0,0,0,0,0,0,0) * VB(0) E.Rank() = 16 Stock = VB(Lambda[1]) + VB(Lambda[3])(-1) + VB(Lambda[1])(-1) Stock.Rank() = 10 VB(Lambda[1]) VB(Lambda[3])(-1) VB(Lambda[1])(-1) X: Smooth projective variety D9/P({1, 3, 4, 5, 6, 7, 8, 9}). n=10 d=29 r=14 E = D9(1,0,0,0,0,0,0,0,0) * VB(0) E.Rank() = 18 Stock = VB(Lambda[1]) + VB(Lambda[3])(-1) + VB(Lambda[1])(-1) Stock.Rank() = 11 VB(Lambda[1]) VB(Lambda[3])(-1) VB(Lambda[1])(-1) X: Smooth projective variety D10/P({1, 3, 4, 5, 6, 7, 8, 9, 10}). n=10 d=33 r=16 E = D10(1,0,0,0,0,0,0,0,0,0) * VB(0) E.Rank() = 20 Stock = VB(Lambda[1]) + VB(Lambda[3])(-1) + VB(Lambda[1])(-1) Stock.Rank() = 12 VB(Lambda[1]) VB(Lambda[3])(-1) VB(Lambda[1])(-1)