CoCalc -- orthogonal_Grassmannian.ipynb

Environment to perform calculations of equivariant vector bundles on homogeneous varieties

Finding exceptional Lefschetz collections

Equivariant_Vector_Bundles_On_Homogeneous_Varieties__0-2 / tests / Weyl_group_coset_representaives_of_minimal_length / orthogonal_Grassmannian.ipynb

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License: GPL3
ubuntu2204

Kernel: SageMath 10.3

In [3]:

%run '/home/user/Equivariant_Vector_Bundles_On_Homogeneous_Varieties__0-0-1/src/Initialize.ipynb'

from IPython.display import clear_output , display , HTML

In [22]:

k = 3
n = 4
N = 2*n+1

X = Orthogonal_Grassmannian(k,N)
print( 'X:' , X )
print()

fw = X.Basis('fw')
Rho = sum(fw.values())
CM = X.Cartan_Matrix( From='ambt' , To='fw' )

Weight1 = 0*fw[1]
cA = X.calU( Weight1 )

cB = dict({})
for Counter , ( WeylElement , ReducedDescription ) in enumerate( X.Weyl_Group_Coset_Representatives_Of_Minimal_Length() , start=1 ) :
    print( 'Counter='+str(Counter) )
    print( 'Weyl group element w ...' )
    print( 3*' ' , '... as matrix:' )
    for Line in str(WeylElement).split('\n') :
        print( 6*' ' , Line )
    print( 3*' ' , '... as reduced word:' , ReducedDescription )
    print( 3*' ' )

    print( 3*' ' , 'It has length len(w)='+str(len(ReducedDescription))+'.' )
    print( 3*' ' )

    Result  = WeylElement.action( Weight1.to_ambient() + Rho.to_ambient() ) - Rho.to_ambient()
    Weight2 = sum([ Coefficient*fw[Node]
                    for Node , Coefficient in enumerate( CM*vector( QQ , [ Result.coefficient(i) for i in range(n) ] ) ,
                                                         start=1
                                                       )
                  ])
    print( 3*' ' , 'w acts on weight '+str(Weight1)+':' )
    print( 6*' ' , Weight1 , '--|w|-->' , Weight2 )
    print( 3*' ' )

    cB.update({ i : X.calU( Weight2 ) })
    print( 3*' ' , 'Consider the irreducible G-equivariant vector bundle cB['+str(i)+'] with highest weight '+str(Weight2)+':' )
    print( 6*' ' , cA.EXT( cB[i] ) )
    print( 3*' ' )
    print()
    print()
    print()

Out[22]:

X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 9-dimensional ambient vector space.

Counter=1
Weyl group element w ...
    ... as matrix:
       [1 0 0 0]
       [0 1 0 0]
       [0 0 1 0]
       [0 0 0 1]
    ... as reduced word: []
   
    It has length len(w)=0.
   
    w acts on weight 0:
       0 --|w|--> 0
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 0:
       {0: B4(0,0,0,0)}
   



Counter=2
Weyl group element w ...
    ... as matrix:
       [1 0 0 0]
       [0 1 0 0]
       [0 0 0 1]
       [0 0 1 0]
    ... as reduced word: [3]
   
    It has length len(w)=1.
   
    w acts on weight 0:
       0 --|w|--> Lambda[2] - 2*Lambda[3] + 2*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[2] - 2*Lambda[3] + 2*Lambda[4]:
       {1: B4(0,0,0,0)}
   



Counter=3
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  1  0  0]
       [ 0  0  0 -1]
       [ 0  0  1  0]
    ... as reduced word: [3, 4]
   
    It has length len(w)=2.
   
    w acts on weight 0:
       0 --|w|--> 2*Lambda[2] - 3*Lambda[3] + 2*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 2*Lambda[2] - 3*Lambda[3] + 2*Lambda[4]:
       {2: B4(0,0,0,0)}
   



Counter=4
Weyl group element w ...
    ... as matrix:
       [1 0 0 0]
       [0 0 1 0]
       [0 0 0 1]
       [0 1 0 0]
    ... as reduced word: [3, 2]
   
    It has length len(w)=2.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] - 3*Lambda[3] + 4*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] - 3*Lambda[3] + 4*Lambda[4]:
       {2: B4(0,0,0,0)}
   



Counter=5
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  0  1  0]
       [ 0  0  0 -1]
       [ 0  1  0  0]
    ... as reduced word: [3, 2, 4]
   
    It has length len(w)=3.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] + Lambda[2] - 4*Lambda[3] + 4*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] + Lambda[2] - 4*Lambda[3] + 4*Lambda[4]:
       {3: B4(0,0,0,0)}
   



Counter=6
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  1  0  0]
       [ 0  0 -1  0]
       [ 0  0  0  1]
    ... as reduced word: [3, 4, 3]
   
    It has length len(w)=3.
   
    w acts on weight 0:
       0 --|w|--> 3*Lambda[2] - 3*Lambda[3]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 3*Lambda[2] - 3*Lambda[3]:
       {3: B4(0,0,0,0)}
   



Counter=7
Weyl group element w ...
    ... as matrix:
       [0 1 0 0]
       [0 0 1 0]
       [0 0 0 1]
       [1 0 0 0]
    ... as reduced word: [3, 2, 1]
   
    It has length len(w)=3.
   
    w acts on weight 0:
       0 --|w|--> -4*Lambda[3] + 6*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight -4*Lambda[3] + 6*Lambda[4]:
       {3: B4(0,0,0,0)}
   



Counter=8
Weyl group element w ...
    ... as matrix:
       [ 0  1  0  0]
       [ 0  0  1  0]
       [ 0  0  0 -1]
       [ 1  0  0  0]
    ... as reduced word: [3, 2, 1, 4]
   
    It has length len(w)=4.
   
    w acts on weight 0:
       0 --|w|--> Lambda[2] - 5*Lambda[3] + 6*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[2] - 5*Lambda[3] + 6*Lambda[4]:
       {4: B4(0,0,0,0)}
   



Counter=9
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  0  0  1]
       [ 0  0 -1  0]
       [ 0  1  0  0]
    ... as reduced word: [3, 2, 4, 3]
   
    It has length len(w)=4.
   
    w acts on weight 0:
       0 --|w|--> 2*Lambda[1] + Lambda[2] - 5*Lambda[3] + 4*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 2*Lambda[1] + Lambda[2] - 5*Lambda[3] + 4*Lambda[4]:
       {4: B4(0,0,0,0)}
   



Counter=10
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  0  1  0]
       [ 0 -1  0  0]
       [ 0  0  0  1]
    ... as reduced word: [3, 4, 3, 2]
   
    It has length len(w)=4.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] + 3*Lambda[2] - 4*Lambda[3]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] + 3*Lambda[2] - 4*Lambda[3]:
       {4: B4(0,0,0,0)}
   



Counter=11
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  0  0 -1]
       [ 0  0 -1  0]
       [ 0  1  0  0]
    ... as reduced word: [3, 2, 4, 3, 4]
   
    It has length len(w)=5.
   
    w acts on weight 0:
       0 --|w|--> 3*Lambda[1] - 5*Lambda[3] + 4*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 3*Lambda[1] - 5*Lambda[3] + 4*Lambda[4]:
       {5: B4(0,0,0,0)}
   



Counter=12
Weyl group element w ...
    ... as matrix:
       [ 0  1  0  0]
       [ 0  0  0  1]
       [ 0  0 -1  0]
       [ 1  0  0  0]
    ... as reduced word: [3, 2, 1, 4, 3]
   
    It has length len(w)=5.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] + Lambda[2] - 6*Lambda[3] + 6*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] + Lambda[2] - 6*Lambda[3] + 6*Lambda[4]:
       {5: B4(0,0,0,0)}
   



Counter=13
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  0  0  1]
       [ 0 -1  0  0]
       [ 0  0  1  0]
    ... as reduced word: [3, 2, 4, 3, 2]
   
    It has length len(w)=5.
   
    w acts on weight 0:
       0 --|w|--> 2*Lambda[1] + 2*Lambda[2] - 5*Lambda[3] + 2*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 2*Lambda[1] + 2*Lambda[2] - 5*Lambda[3] + 2*Lambda[4]:
       {5: B4(0,0,0,0)}
   



Counter=14
Weyl group element w ...
    ... as matrix:
       [ 0  1  0  0]
       [ 0  0  1  0]
       [-1  0  0  0]
       [ 0  0  0  1]
    ... as reduced word: [3, 4, 3, 2, 1]
   
    It has length len(w)=5.
   
    w acts on weight 0:
       0 --|w|--> 4*Lambda[2] - 5*Lambda[3]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 4*Lambda[2] - 5*Lambda[3]:
       {5: B4(0,0,0,0)}
   



Counter=15
Weyl group element w ...
    ... as matrix:
       [ 0  1  0  0]
       [ 0  0  0 -1]
       [ 0  0 -1  0]
       [ 1  0  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 4]
   
    It has length len(w)=6.
   
    w acts on weight 0:
       0 --|w|--> 2*Lambda[1] - 6*Lambda[3] + 6*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 2*Lambda[1] - 6*Lambda[3] + 6*Lambda[4]:
       {6: B4(0,0,0,0)}
   



Counter=16
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  0  0 -1]
       [ 0 -1  0  0]
       [ 0  0  1  0]
    ... as reduced word: [3, 2, 4, 3, 2, 4]
   
    It has length len(w)=6.
   
    w acts on weight 0:
       0 --|w|--> 3*Lambda[1] + Lambda[2] - 5*Lambda[3] + 2*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 3*Lambda[1] + Lambda[2] - 5*Lambda[3] + 2*Lambda[4]:
       {6: B4(0,0,0,0)}
   



Counter=17
Weyl group element w ...
    ... as matrix:
       [ 0  0  1  0]
       [ 0  0  0  1]
       [ 0 -1  0  0]
       [ 1  0  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2]
   
    It has length len(w)=6.
   
    w acts on weight 0:
       0 --|w|--> 2*Lambda[2] - 7*Lambda[3] + 6*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 2*Lambda[2] - 7*Lambda[3] + 6*Lambda[4]:
       {6: B4(0,0,0,0)}
   



Counter=18
Weyl group element w ...
    ... as matrix:
       [ 0  1  0  0]
       [ 0  0  0  1]
       [-1  0  0  0]
       [ 0  0  1  0]
    ... as reduced word: [3, 2, 4, 3, 2, 1]
   
    It has length len(w)=6.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] + 3*Lambda[2] - 6*Lambda[3] + 2*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] + 3*Lambda[2] - 6*Lambda[3] + 2*Lambda[4]:
       {6: B4(0,0,0,0)}
   



Counter=19
Weyl group element w ...
    ... as matrix:
       [ 0  0  1  0]
       [ 0  0  0 -1]
       [ 0 -1  0  0]
       [ 1  0  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 4]
   
    It has length len(w)=7.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] + Lambda[2] - 7*Lambda[3] + 6*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] + Lambda[2] - 7*Lambda[3] + 6*Lambda[4]:
       {7: B4(0,0,0,0)}
   



Counter=20
Weyl group element w ...
    ... as matrix:
       [ 0  1  0  0]
       [ 0  0  0 -1]
       [-1  0  0  0]
       [ 0  0  1  0]
    ... as reduced word: [3, 2, 4, 3, 2, 1, 4]
   
    It has length len(w)=7.
   
    w acts on weight 0:
       0 --|w|--> 2*Lambda[1] + 2*Lambda[2] - 6*Lambda[3] + 2*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 2*Lambda[1] + 2*Lambda[2] - 6*Lambda[3] + 2*Lambda[4]:
       {7: B4(0,0,0,0)}
   



Counter=21
Weyl group element w ...
    ... as matrix:
       [ 1  0  0  0]
       [ 0  0 -1  0]
       [ 0 -1  0  0]
       [ 0  0  0  1]
    ... as reduced word: [3, 2, 4, 3, 2, 4, 3]
   
    It has length len(w)=7.
   
    w acts on weight 0:
       0 --|w|--> 4*Lambda[1] - 4*Lambda[3]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 4*Lambda[1] - 4*Lambda[3]:
       {7: B4(0,0,0,0)}
   



Counter=22
Weyl group element w ...
    ... as matrix:
       [ 0  0  1  0]
       [ 0  0  0  1]
       [-1  0  0  0]
       [ 0  1  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 1]
   
    It has length len(w)=7.
   
    w acts on weight 0:
       0 --|w|--> 3*Lambda[2] - 7*Lambda[3] + 4*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 3*Lambda[2] - 7*Lambda[3] + 4*Lambda[4]:
       {7: B4(0,0,0,0)}
   



Counter=23
Weyl group element w ...
    ... as matrix:
       [ 0  0  1  0]
       [ 0  0  0 -1]
       [-1  0  0  0]
       [ 0  1  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 1, 4]
   
    It has length len(w)=8.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] + 2*Lambda[2] - 7*Lambda[3] + 4*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] + 2*Lambda[2] - 7*Lambda[3] + 4*Lambda[4]:
       {8: B4(0,0,0,0)}
   



Counter=24
Weyl group element w ...
    ... as matrix:
       [ 0  0  0  1]
       [ 0  0 -1  0]
       [ 0 -1  0  0]
       [ 1  0  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 4, 3]
   
    It has length len(w)=8.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] - 7*Lambda[3] + 6*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] - 7*Lambda[3] + 6*Lambda[4]:
       {8: B4(0,0,0,0)}
   



Counter=25
Weyl group element w ...
    ... as matrix:
       [ 0  1  0  0]
       [ 0  0 -1  0]
       [-1  0  0  0]
       [ 0  0  0  1]
    ... as reduced word: [3, 2, 4, 3, 2, 1, 4, 3]
   
    It has length len(w)=8.
   
    w acts on weight 0:
       0 --|w|--> 3*Lambda[1] + Lambda[2] - 5*Lambda[3]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 3*Lambda[1] + Lambda[2] - 5*Lambda[3]:
       {8: B4(0,0,0,0)}
   



Counter=26
Weyl group element w ...
    ... as matrix:
       [ 0  0  0 -1]
       [ 0  0 -1  0]
       [ 0 -1  0  0]
       [ 1  0  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 4, 3, 4]
   
    It has length len(w)=9.
   
    w acts on weight 0:
       0 --|w|--> -7*Lambda[3] + 6*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight -7*Lambda[3] + 6*Lambda[4]:
       {9: B4(0,0,0,0)}
   



Counter=27
Weyl group element w ...
    ... as matrix:
       [ 0  0  0  1]
       [ 0  0 -1  0]
       [-1  0  0  0]
       [ 0  1  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 1, 4, 3]
   
    It has length len(w)=9.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] + Lambda[2] - 7*Lambda[3] + 4*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] + Lambda[2] - 7*Lambda[3] + 4*Lambda[4]:
       {9: B4(0,0,0,0)}
   



Counter=28
Weyl group element w ...
    ... as matrix:
       [ 0  0  1  0]
       [ 0 -1  0  0]
       [-1  0  0  0]
       [ 0  0  0  1]
    ... as reduced word: [3, 2, 4, 3, 2, 1, 4, 3, 2]
   
    It has length len(w)=9.
   
    w acts on weight 0:
       0 --|w|--> 3*Lambda[1] - 5*Lambda[3]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 3*Lambda[1] - 5*Lambda[3]:
       {9: B4(0,0,0,0)}
   



Counter=29
Weyl group element w ...
    ... as matrix:
       [ 0  0  0 -1]
       [ 0  0 -1  0]
       [-1  0  0  0]
       [ 0  1  0  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 1, 4, 3, 4]
   
    It has length len(w)=10.
   
    w acts on weight 0:
       0 --|w|--> Lambda[2] - 7*Lambda[3] + 4*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[2] - 7*Lambda[3] + 4*Lambda[4]:
       {10: B4(0,0,0,0)}
   



Counter=30
Weyl group element w ...
    ... as matrix:
       [ 0  0  0  1]
       [ 0 -1  0  0]
       [-1  0  0  0]
       [ 0  0  1  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 1, 4, 3, 2]
   
    It has length len(w)=10.
   
    w acts on weight 0:
       0 --|w|--> 2*Lambda[1] - 6*Lambda[3] + 2*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight 2*Lambda[1] - 6*Lambda[3] + 2*Lambda[4]:
       {10: B4(0,0,0,0)}
   



Counter=31
Weyl group element w ...
    ... as matrix:
       [ 0  0  0 -1]
       [ 0 -1  0  0]
       [-1  0  0  0]
       [ 0  0  1  0]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 4]
   
    It has length len(w)=11.
   
    w acts on weight 0:
       0 --|w|--> Lambda[1] - 6*Lambda[3] + 2*Lambda[4]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight Lambda[1] - 6*Lambda[3] + 2*Lambda[4]:
       {11: B4(0,0,0,0)}
   



Counter=32
Weyl group element w ...
    ... as matrix:
       [ 0  0 -1  0]
       [ 0 -1  0  0]
       [-1  0  0  0]
       [ 0  0  0  1]
    ... as reduced word: [3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 4, 3]
   
    It has length len(w)=12.
   
    w acts on weight 0:
       0 --|w|--> -5*Lambda[3]
   
    Consider the irreducible G-equivariant vector bundle cB[I] with highest weight -5*Lambda[3]:
       {12: B4(0,0,0,0)}
   

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