Supercrystals#
- class sage.categories.supercrystals.SuperCrystals(s=None)#
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: 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]]]
- 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: 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]]]
- 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: 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)]
- connected_components()#
Return the connected components of
self
as subcrystals.EXAMPLES:
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]]]]
- connected_components_generators()#
Return the tuple of genuine highest weight elements of
self
.EXAMPLES:
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])
- digraph(index_set=None)#
Return the
DiGraph
associated toself
.EXAMPLES:
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)]
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: 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: 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])
- genuine_lowest_weight_vectors()#
Return the tuple of genuine lowest weight elements of
self
.EXAMPLES:
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])
- highest_weight_vectors()#
Return the highest weight vectors of
self
.EXAMPLES:
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])
We give an example from [BKK2000] that has fake highest weight vectors:
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]],)
- lowest_weight_vectors()#
Return the lowest weight vectors of
self
.EXAMPLES:
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]]
We give an example from [BKK2000] that has fake lowest weight vectors:
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]],)
- class ParentMethods#
Bases:
object
- tensor(*crystals, **options)#
Return the tensor product of
self
with the crystalsB
.EXAMPLES:
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
- class TensorProducts(category, *args)#
Bases:
TensorProductsCategory
The category of regular crystals constructed by tensor product of regular crystals.
- extra_super_categories()#
EXAMPLES:
sage: from sage.categories.supercrystals import SuperCrystals sage: SuperCrystals().TensorProducts().extra_super_categories() [Category of super crystals]
- super_categories()#
EXAMPLES:
sage: from sage.categories.supercrystals import SuperCrystals sage: C = SuperCrystals() sage: C.super_categories() [Category of crystals]