Kostant-decomposition.ipynb

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

X = Orthogonal_Grassmannian( 2 , 11 )
n = X.Cartan_Degree()
print( X )
print()

assert X.Is_Adjoint() , \
       'The homogeneous variety X is not adjoint.'

fw = X.Basis('fw')
Rho = sum(fw.values())

r = (X.Dimension()-1)/2
print( 'We have '+str(X.Dimension())+' = dim X = 2*r+1 for r='+str(r)+'.' )
print()

#CM = X.Cartan_Matrix( From='ambt' , To='fw' )

i = 3
print( 'Decompose H^0( G/P , wedge^'+str(i)+' T_{G/P} ):' )
TotalCounter = 0
Restiction = ''
if i <= r :
    for p in [ 0 .. floor(i/2) ] :
        print( 'Konstant´s decomposition for i='+str(i-2*p)+' and j='+str(p)+':' )
        Counter = 0
        for Counter , Weight in enumerate( X.Kostant_Space(i-2*p,p,Restiction) , start=1 ) :
            TotalCounter += 1
            if X.Is_Regular( Weight+Rho ) : Regular = bcolors.OKGREEN+'regular'+bcolors.ENDC
            else                          : Regular = bcolors.FAIL+'singular'+bcolors.ENDC
            print( 3*' ' , str(TotalCounter)+')' , 'Weight '+bcolors.OKBLUE+str(Weight)+bcolors.ENDC+' and its sum with rho is '+Regular+'.' )
        if Counter == 0 : print( 3*' ' , 'None.' )

    for p in [ 0 .. floor(i/2-1/2) ] :
        print( 'Konstant´s decomposition for i='+str(i-2*p-1)+' and j='+str(p+1)+':' )
        Counter = 0
        for Counter , Weight in enumerate( X.Kostant_Space(i-2*p-1,p+1,Restiction) , start=1 ) :
            TotalCounter += 1
            if X.Is_Regular( Weight+Rho ) : Regular = bcolors.OKGREEN+'regular'+bcolors.ENDC
            else                          : Regular = bcolors.WARNING+'singular'+bcolors.ENDC
            print( 3*' ' , str(TotalCounter)+')' , 'Weight '+bcolors.OKBLUE+str(Weight)+bcolors.ENDC+' and its sum with rho is '+Regular+'.' )
        if Counter == 0 : print( 3*' ' , 'None.' )

else :
    for p in [ 0 .. r-ceil(i/2) ] :
        print( 'Konstant´s decomposition for i='+str(i+1+2*p)+' and j='+str(-p-1)+':' )
        Counter = 0
        for Counter , Weight in enumerate( X.Kostant_Space(i+1+2*p,-p-1,Restiction) , start=1 ) :
            TotalCounter += 1
            if X.Is_Regular( Weight+Rho ) : Regular = bcolors.OKGREEN+'regular'+bcolors.ENDC
            else                          : Regular = bcolors.WARNING+'singular'+bcolors.ENDC
            print( 3*' ' , str(TotalCounter)+')' , 'Weight '+bcolors.OKBLUE+str(Weight)+bcolors.ENDC+' and its sum with rho is '+Regular+'.' )
        if Counter == 0 : print( 3*' ' , 'None.' )
    for p in [ 0 .. r-ceil(i/2-1/2) ] :
        print( 'Konstant´s decomposition for i='+str(i+2*p)+' and j='+str(-p-2)+':' )
        Counter = 0
        for Counter , Weight in enumerate( X.Kostant_Space(i+2*p,-p-2,Restiction) , start=1 ) :
            TotalCounter += 1
            if X.Is_Regular( Weight+Rho ) : Regular = bcolors.OKGREEN+'regular'+bcolors.ENDC
            else                          : Regular = bcolors.WARNING+'singular'+bcolors.ENDC
            print( 3*' ' , str(TotalCounter)+')' , 'Weight '+bcolors.OKBLUE+str(Weight)+bcolors.ENDC+' and its sum with rho is '+Regular+'.' )
        if Counter == 0 : print( 3*' ' , 'None.' )

Orthogonal grassmannian variety of 2-dimensional isotropic linear subspaces in a 11-dimensional ambient vector space.

We have 15 = dim X = 2*r+1 for r=7.

Decompose H^0( G/P , wedge^3 T_{G/P} ):
Konstant´s decomposition for i=3 and j=0:
    1) Weight 3*Lambda[1] - Lambda[2] + 2*Lambda[5] and its sum with rho is singular.
    2) Weight Lambda[1] - Lambda[2] + Lambda[3] + Lambda[4] and its sum with rho is singular.
Konstant´s decomposition for i=1 and j=1:
    3) Weight Lambda[1] + Lambda[3] and its sum with rho is regular.
Konstant´s decomposition for i=2 and j=1:
    4) Weight 2*Lambda[1] + Lambda[4] and its sum with rho is regular.
    5) Weight 2*Lambda[3] and its sum with rho is regular.
Konstant´s decomposition for i=0 and j=2:
    6) Weight 2*Lambda[2] and its sum with rho is regular.