Gr-k-2n+1.ipynb

Kernel: SageMath 9.8

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

Minimal_n = 2
Maximal_n = 10

Keys   = [ 'Base_Space' , 'Ist/Soll' , 'Tooltips', 'Numerically_Exceptional' , 'Cell_Coloring' , 'Cell_Bordering' ]
Data   = { Key : [] for Key in Keys }
Header = [ str(Node) for Node in [ 1 .. Maximal_n ] ]

for n in [ Minimal_n .. Maximal_n ] :
    for Key in Data.keys() :
        if Key in [ 'CellColoring' , 'CellBordering' ] : Value = ''
        else : Value = None
        Data[Key] += [ n * [ '?' ] + (Maximal_n-n) * [ Value ] ]

    for k in [ 1 .. n ] : #[ 1 .. CartanDegree ] :
        X = Grassmannian( k , 2*n+1 )
        fw = X.Basis('fw')

        LC = X.Fonarev_Collection()

        Case = { Key : None for Key in Keys }
        # Base space
        Case['Base_Space'] = X.__str__( Output_Style='Short' )
        # Ist/Soll and Cell_Bordering
        Ist  = len(LC)
        Soll = X.K0().rank()
        Case['Ist/Soll'] = str(Ist)+ ' / '+str(Soll)
        if Ist == Soll : Case['Cell_Bordering'] = 'Maximal_Expected_Length'
        else           : Case['Cell_Bordering'] = ''
        # Tooltips
        Case['Tooltips'] = ', '.join([ str(Object) for Object in LC.Starting_Block() ]) + '   '
        # Numerically_Exceptional and Cell_Coloring
        if LC.Is_Numerically_Exceptional() :
            Case['Numerically_Exceptional'] = True
            Case['Cell_Coloring']           = 'Numerically_Exceptional'
        else :
            Case['Numerically_Exceptional'] = False
            Case['Cell_Coloring']           = 'Not_Exceptional'
        # Transfer single case into Data-Pool.
        for Key , Value in Case.items() :
            Data[Key][-1][k-1] = Value
        # Implement dataframe.
        Index = pd.Index( [Minimal_n .. n] , name='n:' )
        Columns = pd.Index( [1 .. Maximal_n] , name='k:')
        DF = {}
        for Key , Rows in Data.items() :
            DF.update({ Key : pd.DataFrame( Rows , index=Index , columns=Columns ) })
        S = DF['Ist/Soll'].style
        S = DF['Ist/Soll'].style\
            .set_caption("Working status for exceptional collections on homogeneous varieties belonging to the Cartan family A.")\
            .set_table_styles( [ { 'selector' : '' , 'props' :  'border-collapse: separate;' } ,
                                 { 'selector' : 'caption' , 'props' : 'caption-side: bottom; font-size:1.3em;' } ,
                                 { 'selector' : '.index_name' , 'props' : 'font-style: italic; color: darkgrey; font-weight:normal;' } ,
                                 { 'selector' : 'th:not(.index_name)' , 'props' : 'background-color: #FFA500; color: white;' } ,
                                 { 'selector' : 'th.col_heading' , 'props' : 'text-align: center;' } ,
                                 { 'selector' : 'td' , 'props' : 'text-align: center; font-weight:normal;' } ,
                                 { 'selector' : '.Maximal_Expected_Length' ,'props' : 'border: 2px dashed green;' } ,
                                 { 'selector' : '.Numerically_Exceptional' ,'props' : 'background-color: #e6ffe6;' } ,
                                 { 'selector' : '.Not_Exceptional' , 'props' : 'background-color: #f28080;' } ,
                                 { 'selector' : 'td:hover' , 'props' : 'background-color: #ffffb3;' }
                               ] ,
                               overwrite=False
                             )\
            .set_tooltips( DF['Tooltips'] ,
                           css_class='pd-tt' ,
                           props='visibility: hidden; position: absolute; z-index: 1; border: 1px solid #000066;'
                                 'background-color: white; color: #000066; font-size: 0.8em;'
                                 'transform: translate(0px, -24px); padding: 0.6em; border-radius: 0.5em;'
                         )\
            .set_td_classes( DF['Cell_Coloring'] )
        clear_output(wait=True)
        display(S)

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
Cell In [4], line 32
     30 Case['Tooltips'] = ', '.join([ str(Object) for Object in LC.Starting_Block() ]) + '   '
     31 # Numerically_Exceptional and Cell_Coloring
---> 32 if LC.Is_Numerically_Exceptional() :
     33     Case['Numerically_Exceptional'] = True
     34     Case['Cell_Coloring']           = 'Numerically_Exceptional'
File /tmp/ipykernel_6636/2856276442.py:374, in Lefschetz_Collection.Is_Numerically_Exceptional(self)
    372 def Is_Numerically_Exceptional ( self ) -> bool :
    373     """Tests if the Lefschetz collection is numerically exceptional."""
--> 374     return self.Is_Exceptional( Test_Numerically=True )
File /tmp/ipykernel_6636/2856276442.py:365, in Lefschetz_Collection.Is_Exceptional(self, Test_Numerically)
    363 if Both_Objects_Are_Simultaneously_Accessible_In_Rows_With_Difference_yDelta :
    364     Obj1 = self.Starting_Block(xPos1)
--> 365     Obj2 = self.Starting_Block(xPos2) * self.Twist()**yDelta
    366     for Obj2_Is_SemiOrthogonal_To_Obj1 in Obj2.Is_SemiOrthogonal_To( Obj1 , Test_Numerically ) :
    367         if not Obj2_Is_SemiOrthogonal_To_Obj1 : return False
File /tmp/ipykernel_6636/3214646350.py:33, in Equivariant_Vector_Bundle.__pow__(self, e)
     30 else : Factor = self
     32 for Counter in (ellipsis_range( Integer(1) ,Ellipsis, abs(e) )) :
---> 33     Product *= Factor
     35 return Product
File /tmp/ipykernel_6636/3978931721.py:41, in Irreducible_Equivariant_Vector_Bundle.__mul__(self, other)
     38 Prd_Twist_HW =    self.Highest_Weight_Supported_Over( Nodes=self.Base_Space().Parabolic_Subgroup().Excluded_Nodes() )                            + other.Highest_Weight_Supported_Over( Nodes=other.Base_Space().Parabolic_Subgroup().Excluded_Nodes() )
     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
     45 def Get_Coefficients_And_Multiplicity_Of_Weyl_Character ( Weyl_Character ) :
File /tmp/ipykernel_6636/3978931721.py:316, in Irreducible_Equivariant_Vector_Bundle.Representation_Supported_Over_Levi_Part(self, Decomposition_Style)
    313 elif Decomposition_Style == 'Coarse' :
    314     # Supported over the total Levi Part
    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() ]) )
    319 else : raise ValueError('The input for the decomposition style is unknown.')
File /tmp/ipykernel_6636/2577564027.py:117, in Parabolic_Subgroup_In_Irreducible_Cartan_Group.Initialize_As_Cartan_Group(self)
    112 def Initialize_As_Cartan_Group ( self ) -> Irreducible_Cartan_Group or Reducible_Cartan_Group :
    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)]
    119     else                    : return Direct_Sum_Of_Cartan_Groups( Components )
File /tmp/ipykernel_6636/2577564027.py:117, in <listcomp>(.0)
    112 def Initialize_As_Cartan_Group ( self ) -> Irreducible_Cartan_Group or Reducible_Cartan_Group :
    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)]
    119     else                    : return Direct_Sum_Of_Cartan_Groups( Components )
File /tmp/ipykernel_6636/2577564027.py:37, in Parabolic_Subgroup_In_Irreducible_Cartan_Group.Cartan_Data(self)
     35 """Returns the Cartan data of each irreducible component of ``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)
   1239 def 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)
    576 ind = 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__()
    318 """
    319 if cls.classcall is not None:
--> 320     return cls.classcall(cls, *args, **kwds)
    321 else:
    322     # Fast version of type.__call__(cls, *args, **kwds)
File /ext/sage/9.8/src/sage/combinat/root_system/cartan_matrix.py:307, in CartanMatrix.__classcall_private__(cls, data, index_set, cartan_type, cartan_type_check, borcherds)
    305         data[(reverse[j], reverse[i])] = -l
    306 else:
--> 307     M = matrix(data)
    308     if borcherds:
    309         if not is_borcherds_cartan_matrix(M):
File /ext/sage/9.8/src/sage/matrix/constructor.pyx:643, in sage.matrix.constructor.matrix()
    641 """
    642 immutable = kwds.pop('immutable', False)
--> 643 M = MatrixArgs(*args, **kwds).matrix()
    644 if immutable:
    645     M.set_immutable()
File /ext/sage/9.8/src/sage/matrix/args.pyx:655, in sage.matrix.args.MatrixArgs.matrix()
    653     True
    654 """
--> 655 self.finalize()
    656 
    657 cdef Matrix M
File /ext/sage/9.8/src/sage/matrix/args.pyx:940, in sage.matrix.args.MatrixArgs.finalize()
    938 if MatrixSpace is None:
    939     from .matrix_space import MatrixSpace
--> 940 self.space = MatrixSpace(self.base, self.nrows, self.ncols,
    941         sparse=self.sparse, **self.kwds)
    942 
File /ext/sage/9.8/src/sage/misc/classcall_metaclass.pyx:320, in sage.misc.classcall_metaclass.ClasscallMetaclass.__call__()
    318 """
    319 if cls.classcall is not None:
--> 320     return cls.classcall(cls, *args, **kwds)
    321 else:
    322     # Fast version of type.__call__(cls, *args, **kwds)
File /ext/sage/9.8/src/sage/matrix/matrix_space.py:513, in MatrixSpace.__classcall__(cls, base_ring, nrows, ncols, sparse, implementation, **kwds)
    409 class MatrixSpace(UniqueRepresentation, Parent):
    410     """
    411     The space of matrices of given size and base ring
    412 
   (...)
    510         True
    511     """
--> 513     @staticmethod
    514     def __classcall__(cls, base_ring, nrows, ncols=None, sparse=False, implementation=None, **kwds):
    515         """
    516         Normalize the arguments to call the ``__init__`` constructor.
    517 
   (...)
    557             False
    558         """
    559         if base_ring not in _Rings:
File src/cysignals/signals.pyx:310, in cysignals.signals.python_check_interrupt()
KeyboardInterrupt: 

Yes = '\033[0;32;47m'+'Yes.'+'\033[0;30;47m'
No  = '\033[0;31;47m'+'No.' +'\033[0;30;47m'
for k in [ 1 .. 10 ] :
    print( 150*'-' )
    print( 'Cartan type A_n/P'+str(k)+':' )
    print()
    for n in [ max([ 2 , k ]) .. 10 ] :
        X = Grassmannian( k , n+1 )
        print( 'Base space X='+X.__str__( Output_Style='Short' )+', i.e. X=A'+str(n)+'/P'+str(k) )
        print( 'We consider the collection of '+'\033[0;34;47m'+'Fonarev'+'\033[0;30;47m'+'.' )

        LC = X.Fonarev_Collection()

        Labelling = []
        Rows = []
        for ObjCounter , Obj in enumerate( LC.Starting_Block() , start=1 ) :
            Label = 'E'+str(ObjCounter)
            Labelling += [ Label ]
            Rows += [ [ Label , str(Obj) ] ]
        print( 'Starting block:' )
        show( table( Rows ) )

        print( 'Grid:' )
        show( LC.Grid( Labelling=Labelling ) )

        if LC.Is_Numerically_Exceptional() : Result1 = True  ; Output1 = Yes
        else                               : Result1 = False ; Output1 = No
        print( 'Is it numerically exceptional?' , Output1 )
        if LC.Has_Maximal_Expected_Length() : Result2 = True ; Output2 = Yes
        else                                : Result2 = False ; Output2 = No
        print( 'Has maximal expected length?' , Output2 )

        print()
        print()
        print()

    print()
    print()
    print()
print()

------------------------------------------------------------------------------------------------------------------------------------------------------
Cartan type A_n/P1:

Base space X=Gr(1;3), i.e. X=A2/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(1;4), i.e. X=A3/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(1;5), i.e. X=A4/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(1;6), i.e. X=A5/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(1;7), i.e. X=A6/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(1;8), i.e. X=A7/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(1;9), i.e. X=A8/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(1;10), i.e. X=A9/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(1;11), i.e. X=A10/P1
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.






------------------------------------------------------------------------------------------------------------------------------------------------------
Cartan type A_n/P2:

Base space X=Gr(2;3), i.e. X=A2/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(2;4), i.e. X=A3/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(2;5), i.e. X=A4/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(2;6), i.e. X=A5/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(2;7), i.e. X=A6/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(2;8), i.e. X=A7/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(2;9), i.e. X=A8/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(2;10), i.e. X=A9/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(2;11), i.e. X=A10/P2
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.






------------------------------------------------------------------------------------------------------------------------------------------------------
Cartan type A_n/P3:

Base space X=Gr(3;4), i.e. X=A3/P3
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(3;5), i.e. X=A4/P3
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(3;6), i.e. X=A5/P3
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(3;7), i.e. X=A6/P3
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(3;8), i.e. X=A7/P3
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(3;9), i.e. X=A8/P3
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(3;10), i.e. X=A9/P3
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(3;11), i.e. X=A10/P3
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.






------------------------------------------------------------------------------------------------------------------------------------------------------
Cartan type A_n/P4:

Base space X=Gr(4;5), i.e. X=A4/P4
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(4;6), i.e. X=A5/P4
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(4;7), i.e. X=A6/P4
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(4;8), i.e. X=A7/P4
We consider the collection of Fonarev.
Starting block:

Grid:

Is it numerically exceptional? Yes.
Has maximal expected length? Yes.



Base space X=Gr(4;9), i.e. X=A8/P4
We consider the collection of Fonarev.
Starting block:

Grid:

WARNING: Some output was deleted.