Playground_for_Complexes.ipynb

Kernel: SageMath 10.3

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

G = Irreducible_Cartan_Group( 'B' , 4 )
WCR = G.Weyl_Character_Ring()

P = G.Maximal_Parabolic_Subgroup(3)
X = G/P
fw = X.Basis('fw')

calQ        = X.calU( fw[1] ).Extend_Equivariantly_By( X.calU( 2*fw[4] )(-1) )
calQ_Wedge2 = X.calU( fw[2] ).Extend_Equivariantly_By( X.calU( fw[1]+2*fw[4] )(-1) ).Extend_Equivariantly_By( X.calU( 2*fw[4] )(-1) ) # Wedge^2 calQ

def A( x:int , y:int=1 ) -> Irreducible_Equivariant_Vector_Bundle :
    assert x in ZZ and 0 <= x , \
           ValueError('The input for ``x`` must be non-negative integer.')
    assert y in ZZ and 0 <= x , \
           ValueError('The input for ``y`` must be non-negative integer.')
    return X.calU( x*fw[1]+y*fw[X.Cartan_Degree()] )(1-y)

E1 = X.calO()
E2 = calQ
E3 = X.calU().Dual()
E4 = calQ_Wedge2.Extend_Equivariantly_By( X.calO() )
E5 = A(0)
E6 = A(1).Extend_Equivariantly_By( A(0) )
E7 = A(2).Extend_Equivariantly_By( A(1) )

V1     =                                            2*G.rmV(fw[4]) + G.rmV(fw[1]+fw[4])
V_calU =                                              G.rmV(fw[4])
V3     =                                              G.rmV(fw[4])
V5     = 2*G.rmV(1) + G.rmV(fw[1]) + G.rmV(fw[2])
V6     =   G.rmV(1) + G.rmV(fw[1])

print( 'Let A0 be the subcategory of D^b(X) generated by E1 , ... , E6.' )
print()

C1 = X.Complex( { 0 : 'Kernel' ,
                  1 : E1.Multiply_By( V1 ) ,
                  2 : X.calU().Multiply_By( V_calU ) + E3.Multiply_By( V3 ) + E5.Multiply_By( V5 ) ,
                  3 : E6.Multiply_By( V6 ) ,
                  4 : E7
                } )
print( 'Complex C1 describing mutation of E7 through subcategory < A0 > ' )
print( C1 )
print()
for Degree , Numerical_Cohomology in C1.Numerical_Cohomology().items() :
    print( 'Degree='+str(Degree) , ':' , Numerical_Cohomology )
print()
print( 'R1 = Semi-simplification of C1_0:')
R1 = C1.SemiSimplification(0)
print( R1 )
print()

C2 = X.Complex( { 0 : 'Kernel' ,
                  1 : E1.Multiply_By( V1 ) ,
                  2 : X.calU().Multiply_By( V_calU ) + E5.Multiply_By( V5 ),
                  3 : E1(1).Multiply_By( V1 ) ,
                  4 : X.calU()(1).Multiply_By( V_calU ) + E3(1).Multiply_By( V3 ) + E5(1).Multiply_By( V5 ) ,
                  5 : E6(1).Multiply_By( V6 ) ,
                  6 : E7(1)
                } )
print( 'Complex C2 describing mutation of E7(1) through subcategory < A0 , A0(1) > ' )
print( C2 )
print()
for Degree , Numerical_Cohomology in C2.Numerical_Cohomology().items() :
    print( 'Degree='+str(Degree) , ':' , Numerical_Cohomology )
print()
print( 'R2 = Semi-simplification of C2_0:')
R2 = C2.SemiSimplification(0)
print( R2 )
print()

print( 'Gram matrix of the residual category:' )
Rows = [ [ R1.Euler_Sum(R1) , R1.Euler_Sum(R2) ] ,
         [ R2.Euler_Sum(R1) , R2.Euler_Sum(R2) ]
       ]
print( table( Rows , frame=True ) )

Let A0 be the subcategory of D^b(X) generated by E1 , ... , E6.

Complex C1 describing mutation of E7 through subcategory < A0 > 
... --d_-2--> 0 --d_-1--> Kernel --d_0--> ( 2*B4(0,0,0,1) + B4(1,0,0,1) ) * VB(0) --d_1--> B4(0,0,0,1) * VB(Lambda[2] - Lambda[3]) + B4(0,0,0,1) * VB(Lambda[1]) + ( 2*B4(0,0,0,0) + B4(1,0,0,0) + B4(0,1,0,0) ) * VB(Lambda[4]) --d_2--> Equivariant extension of ( B4(0,0,0,0) + B4(1,0,0,0) ) * VB(Lambda[1] + Lambda[4]) by ( B4(0,0,0,0) + B4(1,0,0,0) ) * VB(Lambda[4]) --d_3--> Equivariant extension of VB(2*Lambda[1] + Lambda[4]) by VB(Lambda[1] + Lambda[4]) --d_4--> 0 --d_5--> ...

Degree=0 : 0
Degree=1 : 0
Degree=2 : 0
Degree=3 : 0
Degree=4 : 0
Degree=5 : 0

R1 = Semi-simplification of C1_0:
VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(Lambda[1] - Lambda[3] + Lambda[4]) + 2*B4(0,0,0,0) * VB(-Lambda[3] + Lambda[4]) + VB(Lambda[1] - 2*Lambda[3] + 3*Lambda[4]) + VB(-2*Lambda[3] + 3*Lambda[4])

Complex C2 describing mutation of E7(1) through subcategory < A0 , A0(1) > 
... --d_-2--> 0 --d_-1--> Kernel --d_0--> ( 2*B4(0,0,0,1) + B4(1,0,0,1) ) * VB(0) --d_1--> B4(0,0,0,1) * VB(Lambda[2] - Lambda[3]) + ( 2*B4(0,0,0,0) + B4(1,0,0,0) + B4(0,1,0,0) ) * VB(Lambda[4]) --d_2--> ( 2*B4(0,0,0,1) + B4(1,0,0,1) ) * VB(Lambda[3]) --d_3--> B4(0,0,0,1) * VB(Lambda[2]) + B4(0,0,0,1) * VB(Lambda[1] + Lambda[3]) + ( 2*B4(0,0,0,0) + B4(1,0,0,0) + B4(0,1,0,0) ) * VB(Lambda[3] + Lambda[4]) --d_4--> Equivariant extension of ( B4(0,0,0,0) + B4(1,0,0,0) ) * VB(Lambda[1] + Lambda[3] + Lambda[4]) by ( B4(0,0,0,0) + B4(1,0,0,0) ) * VB(Lambda[3] + Lambda[4]) --d_5--> Equivariant extension of VB(2*Lambda[1] + Lambda[3] + Lambda[4]) by VB(Lambda[1] + Lambda[3] + Lambda[4]) --d_6--> 0 --d_7--> ...

Degree=0 : 0
Degree=1 : 0
Degree=2 : 0
Degree=3 : 0
Degree=4 : 0
Degree=5 : 0
Degree=6 : 0
Degree=7 : 0

R2 = Semi-simplification of C2_0:
2*B4(0,0,0,0) * VB(Lambda[1] - Lambda[3] + Lambda[4]) + 2*B4(0,0,0,0) * VB(-Lambda[3] + Lambda[4]) + VB(Lambda[2] - Lambda[3] + Lambda[4]) + VB(2*Lambda[1] - Lambda[3] + Lambda[4]) + VB(Lambda[1] - 2*Lambda[3] + 3*Lambda[4]) + VB(-2*Lambda[3] + 3*Lambda[4])

Gram matrix of the residual category:
+-------------+-------------+
| B4(0,0,0,0) | B4(0,0,0,0) |
+-------------+-------------+
| 0           | B4(0,0,0,0) |
+-------------+-------------+

C = X.Complex( [ 'Kernel' ,
                  E6(4).Multiply_By( G.rmV(fw[4]) ) ,
                  E1(5)
               ] )
print( 'Complex C describing left mutation of E7(1) through subcategory cD = < A0 , A1(1) , ... , A4(4) > ' )
print( C )
print()
for Degree , Numerical_Cohomology in C.Numerical_Cohomology().items() :
    print( 'Degree='+str(Degree) , ':' , Numerical_Cohomology )
print()
print( 'Semi-simplification of C_0:')
print( C.SemiSimplification(0) )

Complex C describing left mutation of E7(1) through subcategory cD = < A0 , A1(1) , ... , A4(4) > 
... --d_-2--> 0 --d_-1--> Kernel --d_0--> Equivariant extension of B4(0,0,0,1) * VB(Lambda[1] + 4*Lambda[3] + Lambda[4]) by B4(0,0,0,1) * VB(4*Lambda[3] + Lambda[4]) --d_1--> VB(5*Lambda[3]) --d_2--> 0 --d_3--> ...

Degree=0 : 0
Degree=1 : 0
Degree=2 : 0
Degree=3 : 0

Semi-simplification of C_0:
VB(5*Lambda[3]) + VB(Lambda[1] + Lambda[2] + 3*Lambda[3] + 2*Lambda[4]) + VB(Lambda[1] + Lambda[2] + 4*Lambda[3]) + 2*B4(0,0,0,0) * VB(4*Lambda[3] + 2*Lambda[4]) + VB(2*Lambda[1] + 3*Lambda[3] + 2*Lambda[4]) + VB(2*Lambda[1] + 4*Lambda[3]) + 2*B4(0,0,0,0) * VB(Lambda[2] + 3*Lambda[3] + 2*Lambda[4]) + 2*B4(0,0,0,0) * VB(Lambda[2] + 4*Lambda[3]) + VB(Lambda[1] + 4*Lambda[3] + 2*Lambda[4]) + VB(Lambda[1] + 5*Lambda[3]) + 2*B4(0,0,0,0) * VB(Lambda[1] + 3*Lambda[3] + 2*Lambda[4]) + 2*B4(0,0,0,0) * VB(Lambda[1] + 4*Lambda[3]) + VB(3*Lambda[3] + 2*Lambda[4]) + VB(4*Lambda[3])

M = E4(4)
Result = X.Parent_Group().Weyl_Character_Ring()(0)

for Degree , Object in C :
    if type(Object) != str :
        Result += (-1)^Degree * M.Euler_Sum(Object)

print( Result )

-2*B4(0,0,0,0) - 2*B4(1,0,0,0) - 3*B4(0,1,0,0) - 2*B4(0,0,1,0) - 2*B4(0,0,0,2) - B4(2,0,0,0) - B4(1,1,0,0) - B4(1,0,1,0) - B4(1,0,0,2)