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Environment to perform calculations of equivariant vector bundles on homogeneous varieties

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License: GPL3
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Kernel: SageMath 10.3
%run '/home/user/Equivariant_Vector_Bundles_On_Homogeneous_Varieties__0-0-1/src/Initialize.ipynb'

k = 3
n = 9

N = 2*n+1
X = Orthogonal_Grassmannian(k,N)
G = X.Parent_Group()
fw = X.Basis('fw')

Output = dict({})
Output.update({ 1 : '' })
Output[1] += 'X: '+str(X)+'\n'
Output[1] += '(n='+str(n)+')'+'\n'
d = X.Dimension()
Output[1] += 3*' '+'Dimension: '+str(d)+'\n'
lMax = X.K0().rank()
Output[1] += 3*' '+'Rank of K0(X) (max. collection length): '+str(lMax)+'\n'
wMax = X.Fano_Index()
Output[1] += 3*' '+'Fano index (max. orbit length): '+str(wMax)+'\n'
Output[1] += '\n'

Output.update({ 2 : '' })
Output[2] += 3*' '+'Consecutive Lefschetz collection.'+'\n'
LC1 = X.My_Collection( Modus='Con' )
l1 = len(LC1)
Output[2] += 3*' '+'Starting block:'+'\n'
w1_0 = len( LC1.Starting_Block() )
s = ceil( math.log10( w1_0 ) )
for i , cE in enumerate ( LC1.Starting_Block() , start=1 ) :
    Output[2] += 6*' '+(s-floor(math.log10(i))-1)*' '+str(i)+' '+str(cE)+'\n'
Output[2] += '\n'

Output.update({ 3 : '' })
Output[3] += 3*' '+'Alternating Lefschetz collection.'+'\n'
LC2 = X.My_Collection( Modus='Alt' )
l2 = len(LC2)
Output[3] += 3*' '+'Starting block:'+'\n'
w2_0 = len( LC2.Starting_Block() )
s = ceil( math.log10( w2_0 ) )
for i , cE in enumerate ( LC2.Starting_Block() , start=1 ) :
    Output[3] += 6*' '+(s-floor(math.log10(i))-1)*' '+str(i)+' '+str(cE)+'\n'
Output[3] += '\n'

Output.update({ 4 : '' })
Output[4] += 3*' '+'Gap.'+'\n'
Output[4] += 6*' '+'       lMax-l = '+str(lMax-l1)+'\n'
Output[4] += 6*' '+'(lMax-l)/wMax = '+str( (lMax-l1)/wMax  )+'\n'
Output[4] += '\n'

Output.update({ 5 : '' })
Output[5] += 3*' '+'Try to fill the gap.'+'\n'
Start_With = 13
if 0 < Start_With :
    Output[5] += 3*' '+'Start later than candidate #'+str(Start_With)+'.'+'\n'
Output[5] += '\n'
Counter = 0

for Partition in IntegerListsLex( length=3 , \
                                  ceiling=(k-1)*[2*(n-k)]+[0] ,
                                  max_slope=0 \
                                ) :

    Counter += 1
    if Start_With < Counter :
        # Present output.
        clear_output(wait=True)
        for Key , Line in Output.items() :
            print(Line)

        print( 3*' ' , 'Current counter: '+bcolors.OKBLUE+str(Counter)+bcolors.ENDC )
        print()

        # Construct candidate.
        Partition = list(Partition)
        Coefficients = [ Partition[Counter-1]-Partition[Counter] for Counter in range( 1 , len(Partition) ) ]
        Candidate = X.calU( sum([ X.Null_Weight() ] + [ Coefficient*fw[Node] for Node , Coefficient in enumerate( Coefficients , start=1 ) ]) )

        # Check if candidate is admissible.
        Candidate_Is_Admissible = False
        if Candidate.Is_Exceptional( Test_Numerically=True ) :
            Candidate_Is_Admissible = True
        if Candidate_Is_Admissible :
            Candidate_Is_Relevant = False
            Possible_Output = ''
            Line = 6*' '+'Test for candidate #'+str(Counter)+':'
            print( Line )
            Possible_Output += Line+'\n'
            Line = 9*' '+'Semi-simplification is '+bcolors.OKBLUE+str(Candidate)+bcolors.ENDC+'.'
            print( Line )
            Possible_Output += Line+'\n'
            Line = ''
            print( Line )
            Possible_Output += Line+'\n'
            for i , LC in enumerate([ LC1 , LC2 ] , start=1 ) :
                Line += 9*' '+'Can candidate be embedded in the Lefschetz collection LC'+str(i)+'?'
                print( Line )
                Possible_Output += Line+'\n'
                for xPos , Admissible_Columns in LC.Test_For_Extension ( New_Object=Candidate , Test_Numerically=True , Relevant_Columns=set([ 33 .. 42]) , Test_If_Self_Is_Exceptional=False ) :
                    if 0 < len(Admissible_Columns) :
                        Candidate_Is_Relevant = True
                        Line = 12*' '+bcolors.OKGREEN+'Yes'+bcolors.ENDC+' after row '+str(xPos)+' to the columns '+bcolors.OKBLUE+str(Admissible_Columns)+bcolors.ENDC+'.'
                        print( Line )
                        Possible_Output += Line+'\n'
                    else :
                        Line = 12*' '+bcolors.FAIL+'Not'+bcolors.ENDC+' after row '+str(xPos)+'.'
                        print( Line )
                        Possible_Output += Line+'\n'
                Line = ''
                print( Line )
                Possible_Output += Line+'\n'

            if Candidate_Is_Relevant :
                Output[5] += Possible_Output
X: Orthogonal grassmannian variety of 3-dimensional isotropic linear subspaces in a 19-dimensional ambient vector space. (n=9) Dimension: 42 Rank of K0(X) (max. collection length): 672 Fano index (max. orbit length): 15 Consecutive Lefschetz collection. Starting block: 1 VB(0) 2 VB(Lambda[1]) 3 VB(Lambda[2]) 4 VB(2*Lambda[1]) 5 VB(Lambda[1] + Lambda[2]) 6 VB(2*Lambda[2]) 7 VB(3*Lambda[1]) 8 VB(2*Lambda[1] + Lambda[2]) 9 VB(Lambda[1] + 2*Lambda[2]) 10 VB(3*Lambda[2]) 11 VB(4*Lambda[1]) 12 VB(3*Lambda[1] + Lambda[2]) 13 VB(2*Lambda[1] + 2*Lambda[2]) 14 VB(Lambda[1] + 3*Lambda[2]) 15 VB(4*Lambda[2]) 16 VB(5*Lambda[1]) 17 VB(4*Lambda[1] + Lambda[2]) 18 VB(3*Lambda[1] + 2*Lambda[2]) 19 VB(2*Lambda[1] + 3*Lambda[2]) 20 VB(Lambda[1] + 4*Lambda[2]) 21 VB(5*Lambda[2]) 22 VB(6*Lambda[1]) 23 VB(5*Lambda[1] + Lambda[2]) 24 VB(4*Lambda[1] + 2*Lambda[2]) 25 VB(3*Lambda[1] + 3*Lambda[2]) 26 VB(2*Lambda[1] + 4*Lambda[2]) 27 VB(Lambda[1] + 5*Lambda[2]) 28 VB(6*Lambda[2]) 29 VB(6*Lambda[1] + Lambda[2]) 30 VB(5*Lambda[1] + 2*Lambda[2]) 31 VB(4*Lambda[1] + 3*Lambda[2]) 32 VB(6*Lambda[1] + 2*Lambda[2]) 33 VB(5*Lambda[1] + 3*Lambda[2]) 34 VB(6*Lambda[1] + 3*Lambda[2]) 35 VB(Lambda[9]) 36 Equivariant extension of VB(Lambda[1] + Lambda[9]) by VB(Lambda[9]) 37 Equivariant extension of VB(2*Lambda[1] + Lambda[9]) by VB(Lambda[1] + Lambda[9]) 38 Equivariant extension of VB(3*Lambda[1] + Lambda[9]) by VB(2*Lambda[1] + Lambda[9]) 39 Equivariant extension of VB(4*Lambda[1] + Lambda[9]) by VB(3*Lambda[1] + Lambda[9]) 40 Equivariant extension of VB(5*Lambda[1] + Lambda[9]) by VB(4*Lambda[1] + Lambda[9]) 41 Equivariant extension of VB(6*Lambda[1] + Lambda[9]) by VB(5*Lambda[1] + Lambda[9]) 42 Equivariant extension of VB(7*Lambda[1] + Lambda[9]) by VB(6*Lambda[1] + Lambda[9]) Alternating Lefschetz collection. Starting block: 1 VB(0) 2 VB(Lambda[9]) 3 VB(Lambda[1]) 4 VB(Lambda[2]) 5 Equivariant extension of VB(Lambda[1] + Lambda[9]) by VB(Lambda[9]) 6 VB(2*Lambda[1]) 7 VB(Lambda[1] + Lambda[2]) 8 VB(2*Lambda[2]) 9 Equivariant extension of VB(2*Lambda[1] + Lambda[9]) by VB(Lambda[1] + Lambda[9]) 10 VB(3*Lambda[1]) 11 VB(2*Lambda[1] + Lambda[2]) 12 VB(Lambda[1] + 2*Lambda[2]) 13 VB(3*Lambda[2]) 14 Equivariant extension of VB(3*Lambda[1] + Lambda[9]) by VB(2*Lambda[1] + Lambda[9]) 15 VB(4*Lambda[1]) 16 VB(3*Lambda[1] + Lambda[2]) 17 VB(2*Lambda[1] + 2*Lambda[2]) 18 VB(Lambda[1] + 3*Lambda[2]) 19 VB(4*Lambda[2]) 20 Equivariant extension of VB(4*Lambda[1] + Lambda[9]) by VB(3*Lambda[1] + Lambda[9]) 21 VB(5*Lambda[1]) 22 VB(4*Lambda[1] + Lambda[2]) 23 VB(3*Lambda[1] + 2*Lambda[2]) 24 VB(2*Lambda[1] + 3*Lambda[2]) 25 VB(Lambda[1] + 4*Lambda[2]) 26 VB(5*Lambda[2]) 27 Equivariant extension of VB(5*Lambda[1] + Lambda[9]) by VB(4*Lambda[1] + Lambda[9]) 28 VB(6*Lambda[1]) 29 VB(5*Lambda[1] + Lambda[2]) 30 VB(4*Lambda[1] + 2*Lambda[2]) 31 VB(3*Lambda[1] + 3*Lambda[2]) 32 VB(2*Lambda[1] + 4*Lambda[2]) 33 VB(Lambda[1] + 5*Lambda[2]) 34 VB(6*Lambda[2]) 35 Equivariant extension of VB(6*Lambda[1] + Lambda[9]) by VB(5*Lambda[1] + Lambda[9]) 36 VB(6*Lambda[1] + Lambda[2]) 37 VB(5*Lambda[1] + 2*Lambda[2]) 38 VB(4*Lambda[1] + 3*Lambda[2]) 39 Equivariant extension of VB(7*Lambda[1] + Lambda[9]) by VB(6*Lambda[1] + Lambda[9]) 40 VB(6*Lambda[1] + 2*Lambda[2]) 41 VB(5*Lambda[1] + 3*Lambda[2]) 42 VB(6*Lambda[1] + 3*Lambda[2]) Gap. lMax-l = 50 (lMax-l)/wMax = 10/3 Try to fill the gap. Start later than candidate #13. Current counter: 24 Test for candidate #24: Semi-simplification is VB(10*Lambda[1] + Lambda[2]). Can candidate be embedded in the Lefschetz collection LC1? Not after row 33. Not after row 34. Not after row 35. 


k = 3
n = 9
X = Orthogonal_Grassmannian( k , 2*n+1 )
fw = X.Basis('fw')
print( 'Base space:' , X.__str__( Output_Style='Short' ) )
print( 'rk K0(X):' , X.K0().rank() )
print( 'Fano index:' , X.Fano_Index() )
print()

h = X.Fano_Index()

# Kuznetsov's spinor bundle filtration
calS = X.calS()
F3 = ( X.calU().Exterior_Power(3) * calS )
F2 = ( X.calU().Exterior_Power(2) * calS ).Extend_Equivariantly_By( F3 )
F1 = ( X.calU().Exterior_Power(1) * calS ).Extend_Equivariantly_By( F2 )
F0 = ( X.calU().Exterior_Power(0) * calS ).Extend_Equivariantly_By( F1 )

LC  = X.Lefschetz_Collection( Starting_Block=[] , Support_Pattern='Trivial' )

# Tautological collection
LC += X.Tautological_Collection()
# ... extension of the tautological collection ...
# Degree = 7
E = X.calU( 6*fw[1]+fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
E = X.calU( 5*fw[1]+2*fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
E = X.calU( 4*fw[1]+3*fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
# Degree = 8
E = X.calU( 6*fw[1]+2*fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
E = X.calU( 5*fw[1]+3*fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
E = X.calU( 4*fw[1]+4*fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
## Degree = 9
#E = X.calU( 6*fw[1]+3*fw[2] )
#LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )

## Mysterious collection
## 1
#LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 3*fw[1]+fw[n-1] )(-1).Extend_Equivariantly_By( X.calU( 2*fw[1]+fw[n-2] )(-1) + X.calU( fw[n-1] )(-1) )] , Support_Pattern=h*[1] )
## 2
#E = X.calU( fw[n] )
#LC += X.Lefschetz_Collection( Starting_Block=[X.calU( fw[1]+2*fw[n] )(-1).Extend_Equivariantly_By( X.calU( 2*fw[n] )(-1) + X.calU( fw[n-1] )(-1) )] , Support_Pattern=h*[1] )
## Note: We extend by X.calU( 2*fw[n] )(-1) + X.calU( fw[n-1] )(-1).
##       However, F3 * X.calU( fw[n] ) = X.calU( 2*fw[n] )(-1) + X.calU( fw[n-1] )(-1) + ... + X.calU( fw[k+1] )(-1) + X.calO()

# Spinor collection
# Degree = 1/2
E = X.calU( fw[n] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
# Degree = 3/2
E = X.calO()(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 5/2
E = X.calU( fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 2*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 7/2
E = X.calU( 2*fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 3*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 9/2
E = X.calU( 3*fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 4*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 11/2
E = X.calU( 4*fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 5*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 13/2
E = X.calU( 5*fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 6*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 15/2
E = X.calU( 6*fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 7*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=7*[1] )

print( 'Objects of starting block:' )
Body = [ [ str(Counter)+':' , str(Object) , str(sum([ 1 for Row in LC.Support_Pattern().values() if Row[Counter-1] == True ])) ]
         for Counter , Object in enumerate( LC.Starting_Block() , start=1 )
       ]
show( table(Body) )
print()

print( 'Number of objects:' , len(LC) )
print( 'LC has maximal expected length?' , LC.Has_Maximal_Expected_Length() )
print()

#print( 'Grid of LC:' )
#show( LC.Grid() )
#print()

#LC_Is_Numercially_Exceptional = LC.Is_Numerically_Exceptional()
#print( 'Is LC numerially exceptional?' , LC_Is_Numercially_Exceptional )
#print()
Base space: OGr(3;19) rk K0(X): 672 Fano index: 15 Objects of starting block:
Number of objects: 622 LC has maximal expected length? False 


Universe = [ X.calU( c1*fw[1]+fw[Node2] )(-1) for c1 in [ 0 .. 7 ] for Node2 in [ k .. n-1 ] ] + [ X.calU( c1*fw[1]+2*fw[n] )(-1) for c1 in [ 0 .. 7 ] ]

History = 'History:'+'\n'

StartWith = 0

Counter = 0
for Subset in Subsets([ 0 .. len(Universe)-1 ] , 2 , submultiset=True ) :
    Counter += 1
    if Counter < StartWith : continue

    clear_output(wait=True)

    print( History )
    print()

    New_Object = X.Zero_Vector_Bundle()
    for Index in Subset :
        New_Object += Universe[Index]

    print( 10*'---' )
    print()
    print( 'Counter: '+str(Counter) )
    print( 'Currently tested object: '+str(New_Object) )
    print()

    #Test for object
    #Subsets of length: 3
    #StartWidth: 5930
    #Currently tested object: VB(-Lambda[3] + Lambda[7]) + VB(Lambda[1] - Lambda[3] + Lambda[8]) + VB(6*Lambda[1])

    Results_For_Single_Objects = []
    for Column , Rows in LC.Test_For_Numerically_Exceptional_Proper_Extension( New_Object=New_Object , Relevant_Columns=set([ 28 .. 35 ]), Test_If_Self_Is_Exceptional=False ) :
        Output = 'Result for column='+str(Column)+': '+str(Rows)
        print( Output )
        if 0 < len(Rows) : Results_For_Single_Objects += [ Output ]
    if 0 < len(Results_For_Single_Objects) :
        History += 'Tested object: '+str(New_Object)+'\n'
        History += '\n'.join(Results_For_Single_Objects)
        History += '\n'

    print()
History: ------------------------------ Counter: 7 Currently tested object: VB(0) + VB(Lambda[1] - Lambda[3] + Lambda[4])
--------------------------------------------------------------------------- KeyboardInterrupt Traceback (most recent call last)
File /tmp/ipykernel_564/4161098233.py:179, in Equivariant_Vector_Bundle.EXT(self, other, *p) 178 try : --> 179 return ( E1.Dual() * E2 ).Cohomology( *p ) 180 except :
File /tmp/ipykernel_564/1189217894.py:97, in Direct_Sum_Of_Equivariant_Vector_Bundles.__mul__(self, other) 96 for Weyl_Character_HW , Weyl_Character_Multiplicity in Multiplicity_1 * Multiplicity_2 : ---> 97 for VB , Mult in ( Vector_Bundle_1 * Vector_Bundle_2 ).SemiSimplification().Summands() : 98 Product += self.__class__( Base_Space=self.Base_Space() , Summands=[ ( VB , Mult * Weyl_Character_Multiplicity * WCR(Weyl_Character_HW) ) ] )
File /tmp/ipykernel_564/3978931721.py:41, in Irreducible_Equivariant_Vector_Bundle.__mul__(self, other) 40 # Product of Levi parts, i.e. product of the representations of highest weights supported over the total Levi part ---> 41 Prd_Levi_Rep = self.Representation_Supported_Over_Levi_Part(Decomposition_Style='Coarse') * other.Representation_Supported_Over_Levi_Part(Decomposition_Style='Coarse') 43 # In the remaining part, we merge the separate products over both the twisting part (excluded nodes) and the Levi part (included nodes) 44 # 1. Define subroutine to get coefficients and multiplicities of a given weyl character
File /tmp/ipykernel_564/3978931721.py:316, in Irreducible_Equivariant_Vector_Bundle.Representation_Supported_Over_Levi_Part(self, Decomposition_Style) 315 WCR = self.Base_Space().Parabolic_Subgroup().Initialize_As_Cartan_Group().Weyl_Character_Ring() --> 316 Basis = self.Base_Space().Parabolic_Subgroup().Initialize_As_Cartan_Group().Cartan_Type().root_system().weight_space().fundamental_weights() 317 return WCR( sum([ Highest_Weight[Node]*Basis[Relabel] for Node , Relabel in Relabelling.items() ]) )
File /tmp/ipykernel_564/2577564027.py:117, in Parabolic_Subgroup_In_Irreducible_Cartan_Group.Initialize_As_Cartan_Group(self) 113 """ 114 Returns the Cartan group associated to the Cartan string of ``self``. 115 Thus, one has access to the methods of the class ``Cartan_Group`` and its subclasses. 116 """ --> 117 Components = [ Irreducible_Cartan_Group( Cartan_Family , Cartan_Degree ) for Cartan_Family , Cartan_Degree in self.Cartan_Data() ] 118 if len(Components) == Integer(1) : return Components[Integer(0)]
File /tmp/ipykernel_564/2577564027.py:117, in <listcomp>(.0) 113 """ 114 Returns the Cartan group associated to the Cartan string of ``self``. 115 Thus, one has access to the methods of the class ``Cartan_Group`` and its subclasses. 116 """ --> 117 Components = [ Irreducible_Cartan_Group( Cartan_Family , Cartan_Degree ) for Cartan_Family , Cartan_Degree in self.Cartan_Data() ] 118 if len(Components) == Integer(1) : return Components[Integer(0)]
File /tmp/ipykernel_564/2577564027.py:37, in Parabolic_Subgroup_In_Irreducible_Cartan_Group.Cartan_Data(self) 36 for Connected_Component in self.Connected_Components_Of_Included_Nodes() : ---> 37 Cartan_Subtype = self.Parent_Group().Cartan_Type().subtype( Connected_Component ) 38 yield Cartan_Subtype.type() , len(Cartan_Subtype.index_set())
File /ext/sage/9.8/src/sage/combinat/root_system/cartan_type.py:1256, in CartanType_abstract.subtype(self, index_set) 1240 """ 1241 Return a subtype of ``self`` given by ``index_set``. 1242 (...) 1254 ['C', 3] 1255 """ -> 1256 return self.cartan_matrix().subtype(index_set).cartan_type()
File /ext/sage/9.8/src/sage/combinat/root_system/cartan_matrix.py:578, in CartanMatrix.subtype(self, index_set) 577 I = [ind.index(i) for i in index_set] --> 578 return CartanMatrix(self.matrix_from_rows_and_columns(I, I), index_set)
File /ext/sage/9.8/src/sage/misc/classcall_metaclass.pyx:320, in sage.misc.classcall_metaclass.ClasscallMetaclass.__call__() 319 if cls.classcall is not None: --> 320 return cls.classcall(cls, *args, **kwds) 321 else:
File /ext/sage/9.8/src/sage/combinat/root_system/cartan_matrix.py:331, in CartanMatrix.__classcall_private__(cls, data, index_set, cartan_type, cartan_type_check, borcherds) 329 # FIXME: We have to initialize the CartanMatrix part separately because 330 # of the __cinit__ of the matrix. We should get rid of this workaround --> 331 mat._CM_init(cartan_type, index_set, cartan_type_check) 332 mat._subdivisions = subdivisions
File /ext/sage/9.8/src/sage/combinat/root_system/cartan_matrix.py:391, in CartanMatrix._CM_init(self, cartan_type, index_set, cartan_type_check) 390 self._cartan_type = None --> 391 cartan_type = find_cartan_type_from_matrix(self) 393 self._cartan_type = cartan_type
File /ext/sage/9.8/src/sage/combinat/root_system/cartan_matrix.py:1149, in find_cartan_type_from_matrix(CM) 1148 ct = CartanType(x) -> 1149 T = DiGraph(ct.dynkin_diagram()) # We need a simple digraph here 1150 iso, match = T.is_isomorphic(S, certificate=True, edge_labels=True)
File /ext/sage/9.8/src/sage/graphs/digraph.py:775, in DiGraph.__init__(self, data, pos, loops, format, weighted, data_structure, vertex_labels, name, multiedges, convert_empty_dict_labels_to_None, sparse, immutable) 774 self.set_vertices(data.get_vertices()) --> 775 data._backend.subgraph_given_vertices(self._backend, data) 776 self.name(data.name())
File src/cysignals/signals.pyx:310, in cysignals.signals.python_check_interrupt()
KeyboardInterrupt: During handling of the above exception, another exception occurred: ValueError Traceback (most recent call last) Cell In [23], line 33 27 #Test for object 28 #Subsets of length: 3 29 #StartWidth: 5930 30 #Currently tested object: VB(-Lambda[3] + Lambda[7]) + VB(Lambda[1] - Lambda[3] + Lambda[8]) + VB(6*Lambda[1]) 32 Results_For_Single_Objects = [] ---> 33 for Column , Rows in LC.Test_For_Numerically_Exceptional_Proper_Extension( New_Object=New_Object , Relevant_Columns=set((ellipsis_range( Integer(28) ,Ellipsis, Integer(35) ))), Test_If_Self_Is_Exceptional=False ) : 34 Output = 'Result for column='+str(Column)+': '+str(Rows) 35 print( Output )
File /tmp/ipykernel_564/895810085.py:579, in Lefschetz_Collection.Test_For_Numerically_Exceptional_Proper_Extension(self, New_Object, Relevant_Columns, Test_If_Self_Is_Exceptional) 575 return True 577 for Tested_xPos in Relevant_Columns : 578 #Already_Tested = dict({}) --> 579 yield ( Tested_xPos , [ Tested_yPos for Tested_yPos in range(len(New_Orbit)) if Test_For( Tested_xPos , Tested_yPos ) ] )
File /tmp/ipykernel_564/895810085.py:579, in <listcomp>(.0) 575 return True 577 for Tested_xPos in Relevant_Columns : 578 #Already_Tested = dict({}) --> 579 yield ( Tested_xPos , [ Tested_yPos for Tested_yPos in range(len(New_Orbit)) if Test_For( Tested_xPos , Tested_yPos ) ] )
File /tmp/ipykernel_564/895810085.py:574, in Lefschetz_Collection.Test_For_Numerically_Exceptional_Proper_Extension.<locals>.Test_For(xPos1, yPos1) 570 if not next( Object1.Is_SemiOrthogonal_To( Object2 , Test_Numerically=Test_Numerically ) ) : return False 571 else : 572 #if not yPos2-yPos1 in Already_Tested[xPos2][1] : 573 # Already_Tested[xPos2][1].add( yPos2-yPos1 ) --> 574 if not next( Object2.Is_SemiOrthogonal_To( Object1 , Test_Numerically=Test_Numerically ) ) : return False 575 return True
File /tmp/ipykernel_564/4161098233.py:270, in Equivariant_Vector_Bundle.Is_SemiOrthogonal_To(self, others, Test_Numerically) 267 else : yield False 269 elif Test_Numerically == True : --> 270 if self.Euler_Sum( other ) == WCR(Integer(0)) : yield True 271 else : yield False 273 else :
File /tmp/ipykernel_564/4161098233.py:160, in Equivariant_Vector_Bundle.Euler_Sum(self, other) 158 """Returns sum_{ p } (-1)^p EXT^p(``self``,``other``).""" 159 WCR = self.Base_Space().Parent_Group().Weyl_Character_Ring() --> 160 Result = sum([ (-Integer(1))**Degree * Weyl_Character for Degree , Weyl_Character in self.EXT(other).items() ]) 161 if Result == Integer(0) : return WCR(Integer(0)) 162 else : return Result
File /tmp/ipykernel_564/4161098233.py:181, in Equivariant_Vector_Bundle.EXT(self, other, *p) 179 return ( E1.Dual() * E2 ).Cohomology( *p ) 180 except : --> 181 raise ValueError( 'Not able to compute EXT(E1,E2) with\n'+'E1='+str(E1)+'\n'+'E2='+str(E2)+'.' )
ValueError: Not able to compute EXT(E1,E2) with E1=VB(2*Lambda[2] + 2*Lambda[3]) E2=VB(0) + VB(Lambda[1] - Lambda[3] + Lambda[4]).

Counter: 1 New object which is tested: Equivariant extension of VB(Lambda[1] - Lambda[3] + 2Lambda[9]) by VB(-Lambda[3] + 2Lambda[9]) + VB(-Lambda[3] + Lambda[8]) In the column=28: [] In the column=29: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] In the column=30: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] In the column=31: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] In the column=32: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] In the column=33: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] In the column=34: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] In the column=35: []

Counter: 2 New object which is tested: Equivariant [3, 1]-extension of VB(3Lambda[1] - Lambda[3] + Lambda[8]) by VB(2Lambda[1] - Lambda[3] + Lambda[7]) + VB(-Lambda[3] + Lambda[8]) In the column=28: [] In the column=29: [] In the column=30: [] In the column=31: [] In the column=32: [] In the column=33: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] In the column=34: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] In the column=35: []



Universe = dict({})

for p in range( 0 , n , 2 ) :
    G = X.Parent_Group()
    Dict = {}
    Dict.update({ i : X.calU().Symmetric_Power(-i-1).Multiply_By( G.rmV( fw[1] ).exterior_power(i+1+p) ) for i in [ -p-1 .. -1 ] })
    Dict.update({ 0 : 'Cokernel' })
    C = X.Complex(Dict)
    Universe.update({ p : [ Summand for Summand in C.SemiSimplification(0) ] })

    print( 'Summands of Wedge^'+str(p)+' calQ:' )
    print(', '.join([ str(Summand) for Summand in Universe[p] ]))
    print()
Summands of Wedge^0 calQ: VB(0) Summands of Wedge^2 calQ: VB(-Lambda[3] + Lambda[5]), VB(Lambda[1] - Lambda[3] + Lambda[4]), VB(Lambda[2]) Summands of Wedge^4 calQ: VB(Lambda[1] - Lambda[3] + Lambda[6]), VB(-Lambda[3] + Lambda[7]), VB(Lambda[2] - Lambda[3] + Lambda[5]), VB(Lambda[4]) Summands of Wedge^6 calQ: VB(Lambda[1] - Lambda[3] + Lambda[8]), VB(Lambda[2] - Lambda[3] + Lambda[7]), VB(-Lambda[3] + 2*Lambda[9]), VB(Lambda[6]) Summands of Wedge^8 calQ: VB(Lambda[2] - Lambda[3] + 2*Lambda[9]), VB(Lambda[1] - Lambda[3] + 2*Lambda[9]), VB(Lambda[8]), VB(-Lambda[3] + Lambda[8]) 


def A ( c , Node ) :
    if Node in [ k .. n-1 ] : return X.calU( c*fw[1]+fw[Node] )(-1)
    elif Node in [ n ]      : return X.calU( c*fw[1]+2*fw[n] )(-1)

#E = A(1,4) + A(0,5)+ A(0,3)
E = A(1,6) + A(0,7)+ A(0,5)
print( 'E:' , E )
print( 'len(E.MNEO()):' , len( E.Maximal_Numerically_Exceptional_Orbit() ) )
E: VB(Lambda[1] - Lambda[3] + Lambda[6]) + VB(-Lambda[3] + Lambda[7]) + VB(-Lambda[3] + Lambda[5]) len(E.MNEO()): 15