Supercrystals

class sage.categories.supercrystals.SuperCrystals

Bases: Category_singleton

class Finite(base_category)

Bases: CategoryWithAxiom_singleton

class ElementMethods

Bases: object

is_genuine_highest_weight(index_set=None)

Return whether self is a genuine highest weight element.

INPUT:

• index_set – (optional) the index set of the (sub)crystal on which to check

EXAMPLES:

Sage

sage: B = crystals.Tableaux(['A', [1,1]], shape=[3,2,1])
sage: for b in B.highest_weight_vectors():
....:     print("{} {}".format(b, b.is_genuine_highest_weight()))
[[-2, -2, -2], [-1, -1], [1]] True
[[-2, -2, -2], [-1, 2], [1]] False
[[-2, -2, 2], [-1, -1], [1]] False
sage: [b for b in B if b.is_genuine_highest_weight([-1,0])]
[[[-2, -2, -2], [-1, -1], [1]],
 [[-2, -2, -2], [-1, -1], [2]],
 [[-2, -2, -2], [-1, 2], [2]],
 [[-2, -2, 2], [-1, -1], [2]],
 [[-2, -2, 2], [-1, 2], [2]],
 [[-2, -2, -2], [-1, 2], [1]],
 [[-2, -2, 2], [-1, -1], [1]],
 [[-2, -2, 2], [-1, 2], [1]]]

Python

>>> from sage.all import *
>>> B = crystals.Tableaux(['A', [Integer(1),Integer(1)]], shape=[Integer(3),Integer(2),Integer(1)])
>>> for b in B.highest_weight_vectors():
...     print("{} {}".format(b, b.is_genuine_highest_weight()))
[[-2, -2, -2], [-1, -1], [1]] True
[[-2, -2, -2], [-1, 2], [1]] False
[[-2, -2, 2], [-1, -1], [1]] False
>>> [b for b in B if b.is_genuine_highest_weight([-Integer(1),Integer(0)])]
[[[-2, -2, -2], [-1, -1], [1]],
 [[-2, -2, -2], [-1, -1], [2]],
 [[-2, -2, -2], [-1, 2], [2]],
 [[-2, -2, 2], [-1, -1], [2]],
 [[-2, -2, 2], [-1, 2], [2]],
 [[-2, -2, -2], [-1, 2], [1]],
 [[-2, -2, 2], [-1, -1], [1]],
 [[-2, -2, 2], [-1, 2], [1]]]

is_genuine_lowest_weight(index_set=None)

Return whether self is a genuine lowest weight element.

INPUT:

• index_set – (optional) the index set of the (sub)crystal on which to check

EXAMPLES:

Sage

sage: B = crystals.Tableaux(['A', [1,1]], shape=[3,2,1])
sage: for b in sorted(B.lowest_weight_vectors()):
....:     print("{} {}".format(b, b.is_genuine_lowest_weight()))
[[-2, 1, 2], [-1, 2], [1]] False
[[-2, 1, 2], [-1, 2], [2]] False
[[-1, 1, 2], [1, 2], [2]] True
sage: [b for b in B if b.is_genuine_lowest_weight([-1,0])]
[[[-2, -1, 1], [-1, 1], [1]],
 [[-2, -1, 1], [-1, 1], [2]],
 [[-2, 1, 2], [-1, 1], [2]],
 [[-2, 1, 2], [-1, 1], [1]],
 [[-1, -1, 1], [1, 2], [2]],
 [[-1, -1, 1], [1, 2], [1]],
 [[-1, 1, 2], [1, 2], [2]],
 [[-1, 1, 2], [1, 2], [1]]]

Python

>>> from sage.all import *
>>> B = crystals.Tableaux(['A', [Integer(1),Integer(1)]], shape=[Integer(3),Integer(2),Integer(1)])
>>> for b in sorted(B.lowest_weight_vectors()):
...     print("{} {}".format(b, b.is_genuine_lowest_weight()))
[[-2, 1, 2], [-1, 2], [1]] False
[[-2, 1, 2], [-1, 2], [2]] False
[[-1, 1, 2], [1, 2], [2]] True
>>> [b for b in B if b.is_genuine_lowest_weight([-Integer(1),Integer(0)])]
[[[-2, -1, 1], [-1, 1], [1]],
 [[-2, -1, 1], [-1, 1], [2]],
 [[-2, 1, 2], [-1, 1], [2]],
 [[-2, 1, 2], [-1, 1], [1]],
 [[-1, -1, 1], [1, 2], [2]],
 [[-1, -1, 1], [1, 2], [1]],
 [[-1, 1, 2], [1, 2], [2]],
 [[-1, 1, 2], [1, 2], [1]]]

class ParentMethods

Bases: object

character()

Return the character of self.

Todo

Once the \(WeylCharacterRing\) is implemented, make this consistent with the implementation in sage.categories.classical_crystals.ClassicalCrystals.ParentMethods.character().

EXAMPLES:

Sage

sage: B = crystals.Letters(['A',[1,2]])
sage: B.character()
B[(1, 0, 0, 0, 0)] + B[(0, 1, 0, 0, 0)] + B[(0, 0, 1, 0, 0)]
 + B[(0, 0, 0, 1, 0)] + B[(0, 0, 0, 0, 1)]

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A',[Integer(1),Integer(2)]])
>>> B.character()
B[(1, 0, 0, 0, 0)] + B[(0, 1, 0, 0, 0)] + B[(0, 0, 1, 0, 0)]
 + B[(0, 0, 0, 1, 0)] + B[(0, 0, 0, 0, 1)]

connected_components()

Return the connected components of self as subcrystals.

EXAMPLES:

Sage

sage: B = crystals.Letters(['A', [1,2]])
sage: B.connected_components()
[Subcrystal of The crystal of letters for type ['A', [1, 2]]]

sage: T = B.tensor(B)
sage: T.connected_components()
[Subcrystal of Full tensor product of the crystals
  [The crystal of letters for type ['A', [1, 2]],
   The crystal of letters for type ['A', [1, 2]]],
 Subcrystal of Full tensor product of the crystals
  [The crystal of letters for type ['A', [1, 2]],
   The crystal of letters for type ['A', [1, 2]]]]

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A', [Integer(1),Integer(2)]])
>>> B.connected_components()
[Subcrystal of The crystal of letters for type ['A', [1, 2]]]

>>> T = B.tensor(B)
>>> T.connected_components()
[Subcrystal of Full tensor product of the crystals
  [The crystal of letters for type ['A', [1, 2]],
   The crystal of letters for type ['A', [1, 2]]],
 Subcrystal of Full tensor product of the crystals
  [The crystal of letters for type ['A', [1, 2]],
   The crystal of letters for type ['A', [1, 2]]]]

connected_components_generators()

Return the tuple of genuine highest weight elements of self.

EXAMPLES:

Sage

sage: B = crystals.Letters(['A', [1,2]])
sage: B.genuine_highest_weight_vectors()
(-2,)

sage: T = B.tensor(B)
sage: T.genuine_highest_weight_vectors()
([-2, -1], [-2, -2])
sage: s1, s2 = T.connected_components()
sage: s = s1 + s2
sage: s.genuine_highest_weight_vectors()
([-2, -1], [-2, -2])

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A', [Integer(1),Integer(2)]])
>>> B.genuine_highest_weight_vectors()
(-2,)

>>> T = B.tensor(B)
>>> T.genuine_highest_weight_vectors()
([-2, -1], [-2, -2])
>>> s1, s2 = T.connected_components()
>>> s = s1 + s2
>>> s.genuine_highest_weight_vectors()
([-2, -1], [-2, -2])

digraph(index_set=None)

Return the DiGraph associated to self.

EXAMPLES:

Sage

sage: B = crystals.Letters(['A', [1,3]])
sage: G = B.digraph(); G
Multi-digraph on 6 vertices
sage: Q = crystals.Letters(['Q',3])
sage: G = Q.digraph(); G
Multi-digraph on 3 vertices
sage: G.edges(sort=True)
[(1, 2, -1), (1, 2, 1), (2, 3, -2), (2, 3, 2)]

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A', [Integer(1),Integer(3)]])
>>> G = B.digraph(); G
Multi-digraph on 6 vertices
>>> Q = crystals.Letters(['Q',Integer(3)])
>>> G = Q.digraph(); G
Multi-digraph on 3 vertices
>>> G.edges(sort=True)
[(1, 2, -1), (1, 2, 1), (2, 3, -2), (2, 3, 2)]

The edges of the crystal graph are by default colored using blue for edge 1, red for edge 2, green for edge 3, and dashed with the corresponding color for barred edges. Edge 0 is dotted black:

Sage

sage: view(G)  # optional - dot2tex graphviz, not tested (opens external window)

Python

>>> from sage.all import *
>>> view(G)  # optional - dot2tex graphviz, not tested (opens external window)

genuine_highest_weight_vectors()

Return the tuple of genuine highest weight elements of self.

EXAMPLES:

Sage

sage: B = crystals.Letters(['A', [1,2]])
sage: B.genuine_highest_weight_vectors()
(-2,)

sage: T = B.tensor(B)
sage: T.genuine_highest_weight_vectors()
([-2, -1], [-2, -2])
sage: s1, s2 = T.connected_components()
sage: s = s1 + s2
sage: s.genuine_highest_weight_vectors()
([-2, -1], [-2, -2])

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A', [Integer(1),Integer(2)]])
>>> B.genuine_highest_weight_vectors()
(-2,)

>>> T = B.tensor(B)
>>> T.genuine_highest_weight_vectors()
([-2, -1], [-2, -2])
>>> s1, s2 = T.connected_components()
>>> s = s1 + s2
>>> s.genuine_highest_weight_vectors()
([-2, -1], [-2, -2])

genuine_lowest_weight_vectors()

Return the tuple of genuine lowest weight elements of self.

EXAMPLES:

Sage

sage: B = crystals.Letters(['A', [1,2]])
sage: B.genuine_lowest_weight_vectors()
(3,)

sage: T = B.tensor(B)
sage: T.genuine_lowest_weight_vectors()
([3, 3], [3, 2])
sage: s1, s2 = T.connected_components()
sage: s = s1 + s2
sage: s.genuine_lowest_weight_vectors()
([3, 3], [3, 2])

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A', [Integer(1),Integer(2)]])
>>> B.genuine_lowest_weight_vectors()
(3,)

>>> T = B.tensor(B)
>>> T.genuine_lowest_weight_vectors()
([3, 3], [3, 2])
>>> s1, s2 = T.connected_components()
>>> s = s1 + s2
>>> s.genuine_lowest_weight_vectors()
([3, 3], [3, 2])

highest_weight_vectors()

Return the highest weight vectors of self.

EXAMPLES:

Sage

sage: B = crystals.Letters(['A', [1,2]])
sage: B.highest_weight_vectors()
(-2,)

sage: T = B.tensor(B)
sage: T.highest_weight_vectors()
([-2, -2], [-2, -1])

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A', [Integer(1),Integer(2)]])
>>> B.highest_weight_vectors()
(-2,)

>>> T = B.tensor(B)
>>> T.highest_weight_vectors()
([-2, -2], [-2, -1])

We give an example from [BKK2000] that has fake highest weight vectors:

Sage

sage: B = crystals.Tableaux(['A', [1,1]], shape=[3,2,1])
sage: B.highest_weight_vectors()
([[-2, -2, -2], [-1, -1], [1]],
 [[-2, -2, -2], [-1, 2], [1]],
 [[-2, -2, 2], [-1, -1], [1]])
sage: B.genuine_highest_weight_vectors()
([[-2, -2, -2], [-1, -1], [1]],)

Python

>>> from sage.all import *
>>> B = crystals.Tableaux(['A', [Integer(1),Integer(1)]], shape=[Integer(3),Integer(2),Integer(1)])
>>> B.highest_weight_vectors()
([[-2, -2, -2], [-1, -1], [1]],
 [[-2, -2, -2], [-1, 2], [1]],
 [[-2, -2, 2], [-1, -1], [1]])
>>> B.genuine_highest_weight_vectors()
([[-2, -2, -2], [-1, -1], [1]],)

lowest_weight_vectors()

Return the lowest weight vectors of self.

EXAMPLES:

Sage

sage: B = crystals.Letters(['A', [1,2]])
sage: B.lowest_weight_vectors()
(3,)

sage: T = B.tensor(B)
sage: sorted(T.lowest_weight_vectors())
[[3, 2], [3, 3]]

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A', [Integer(1),Integer(2)]])
>>> B.lowest_weight_vectors()
(3,)

>>> T = B.tensor(B)
>>> sorted(T.lowest_weight_vectors())
[[3, 2], [3, 3]]

We give an example from [BKK2000] that has fake lowest weight vectors:

Sage

sage: B = crystals.Tableaux(['A', [1,1]], shape=[3,2,1])
sage: sorted(B.lowest_weight_vectors())
[[[-2, 1, 2], [-1, 2], [1]],
 [[-2, 1, 2], [-1, 2], [2]],
 [[-1, 1, 2], [1, 2], [2]]]
sage: B.genuine_lowest_weight_vectors()
([[-1, 1, 2], [1, 2], [2]],)

Python

>>> from sage.all import *
>>> B = crystals.Tableaux(['A', [Integer(1),Integer(1)]], shape=[Integer(3),Integer(2),Integer(1)])
>>> sorted(B.lowest_weight_vectors())
[[[-2, 1, 2], [-1, 2], [1]],
 [[-2, 1, 2], [-1, 2], [2]],
 [[-1, 1, 2], [1, 2], [2]]]
>>> B.genuine_lowest_weight_vectors()
([[-1, 1, 2], [1, 2], [2]],)

class ParentMethods

Bases: object

tensor(*crystals, **options)

Return the tensor product of self with the crystals B.

EXAMPLES:

Sage

sage: B = crystals.Letters(['A',[1,2]])
sage: C = crystals.Tableaux(['A',[1,2]], shape = [2,1])
sage: T = C.tensor(B); T
Full tensor product of the crystals
 [Crystal of BKK tableaux of shape [2, 1] of gl(2|3),
  The crystal of letters for type ['A', [1, 2]]]
sage: S = B.tensor(C); S
Full tensor product of the crystals
 [The crystal of letters for type ['A', [1, 2]],
  Crystal of BKK tableaux of shape [2, 1] of gl(2|3)]
sage: G = T.digraph()
sage: H = S.digraph()
sage: G.is_isomorphic(H, edge_labels= True)
True

Python

>>> from sage.all import *
>>> B = crystals.Letters(['A',[Integer(1),Integer(2)]])
>>> C = crystals.Tableaux(['A',[Integer(1),Integer(2)]], shape = [Integer(2),Integer(1)])
>>> T = C.tensor(B); T
Full tensor product of the crystals
 [Crystal of BKK tableaux of shape [2, 1] of gl(2|3),
  The crystal of letters for type ['A', [1, 2]]]
>>> S = B.tensor(C); S
Full tensor product of the crystals
 [The crystal of letters for type ['A', [1, 2]],
  Crystal of BKK tableaux of shape [2, 1] of gl(2|3)]
>>> G = T.digraph()
>>> H = S.digraph()
>>> G.is_isomorphic(H, edge_labels= True)
True

class TensorProducts(category, *args)

Bases: TensorProductsCategory

The category of regular crystals constructed by tensor product of regular crystals.

extra_super_categories()

EXAMPLES:

Sage

sage: from sage.categories.supercrystals import SuperCrystals
sage: SuperCrystals().TensorProducts().extra_super_categories()
[Category of super crystals]

Python

>>> from sage.all import *
>>> from sage.categories.supercrystals import SuperCrystals
>>> SuperCrystals().TensorProducts().extra_super_categories()
[Category of super crystals]

super_categories()

EXAMPLES:

Sage

sage: from sage.categories.supercrystals import SuperCrystals
sage: C = SuperCrystals()
sage: C.super_categories()
[Category of crystals]

Python

>>> from sage.all import *
>>> from sage.categories.supercrystals import SuperCrystals
>>> C = SuperCrystals()
>>> C.super_categories()
[Category of crystals]