decomposition_for_adjoint_generalised_Grassmannians.ipynb

Kernel: SageMath 10.3

%run '/home/user/Hochschild-Kostant-Rosenberg_decomposition/src/initialize.ipynb'

from HochschildKostantRosenberg_decomposition.base_space import *

# Initialise base space.
G = irreducible_Cartan_group( 'B' , 4 )
P = G.parabolic_subgroup(2)
X = G/P
dimX = X.dimension()
r = (dimX-1)/2
print( 'Base space: '+str(X) )
print( 'We have dim X = '+str(dimX)+' = 2*r+1 where r='+bcolors.OKBLUE+str(r)+bcolors.ENDC+'.' )
print()

# Input
i = 3
print( 'Decompose HH^'+bcolors.OKBLUE+str(i)+bcolors.ENDC+'( X ) = H^0( X , wedge^'+str(i)+' T_X ) by Hochschild–Kostant–Rosenberg. ...' )
print()

# Test if given base space is adjoint.
assert X.is_adjoint() , \
       'The homogeneous variety X is not adjoint.'

# Set up necessary data.
fw  = dict( X._parabolic_subgroup._parent_group.Cartan_type().root_system().weight_space().fundamental_weights() )
rho = sum(list(fw.values())) # sum of fundamental weights
total_counter = 0
restiction = ''

# Set up stock of indices.
stock = []
if i <= r :
    stock = [ ( i-2*p   , p    ) for p in [ 0 .. floor(i/2) ]       ] + \
            [ ( i-2*p-1 , p+1  ) for p in [ 0 .. floor(i/2-1/2) ]   ]
else :
    stock = [ ( i+1+2*p , -p-1 ) for p in [ 0 .. floor(r-i/2) ]     ] + \
            [ ( i+2*p   , -p-2 ) for p in [ 0 .. floor(r-i/2+1/2) ] ]

# Compute Kostant space and present output.
print( 'H^0( X , wedge^'+str(i)+' T_X ) consists of the following Kostant spaces:' )
for u , v in stock :
    print( bcolors.OKBLUE+'K( '+str(G)+' , '+str(P)+' , '+str(u)+' , '+str(v)+' )'+bcolors.ENDC+'' )
    counter = 0
    for counter , weight in enumerate( X.Kostant_space(u,v,restiction) , start=1 ) :
        total_counter += 1
        if X.is_regular( weight+rho ) : regular = bcolors.OKGREEN+'regular'+bcolors.ENDC
        else                          : regular = bcolors.FAIL+'singular'+bcolors.ENDC
        string = 3*' '+str(total_counter)+')'
        print( string          , 'V^{ lambda } with lambda = '+bcolors.OKBLUE+str(weight)+bcolors.ENDC )
        print( len(string)*' ' , 'Note: lambda + rho = '+str( weight+rho )+' is '+regular+'.' )
    if counter == 0 : print( 3*' ' , 'None.' )
    print()

Base space: B_4/P_2
We have dim X = 11 = 2*r+1 where r=5.

Decompose HH^3( X ) = H^0( X , wedge^3 T_X ) by Hochschild–Kostant–Rosenberg. ...

H^0( X , wedge^3 T_X ) consists of the following Kostant spaces:
K( B_4 , P_2 , 3 , 0 )
   1) V^{ lambda } with lambda = Lambda[1] - Lambda[2] + Lambda[3] + 2*Lambda[4]
      Note: lambda + rho = 2*Lambda[1] + 2*Lambda[3] + 3*Lambda[4] is singular.
   2) V^{ lambda } with lambda = 3*Lambda[1] - Lambda[2] + 2*Lambda[4]
      Note: lambda + rho = 4*Lambda[1] + Lambda[3] + 3*Lambda[4] is singular.

K( B_4 , P_2 , 1 , 1 )
   3) V^{ lambda } with lambda = Lambda[1] + Lambda[3]
      Note: lambda + rho = 2*Lambda[1] + Lambda[2] + 2*Lambda[3] + Lambda[4] is regular.

K( B_4 , P_2 , 2 , 1 )
   4) V^{ lambda } with lambda = 2*Lambda[1] + 2*Lambda[4]
      Note: lambda + rho = 3*Lambda[1] + Lambda[2] + Lambda[3] + 3*Lambda[4] is regular.
   5) V^{ lambda } with lambda = 2*Lambda[3]
      Note: lambda + rho = Lambda[1] + Lambda[2] + 3*Lambda[3] + Lambda[4] is regular.

K( B_4 , P_2 , 0 , 2 )
   6) V^{ lambda } with lambda = 2*Lambda[2]
      Note: lambda + rho = Lambda[1] + 3*Lambda[2] + Lambda[3] + Lambda[4] is regular.